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Various candidates would be:

self-referential sentences such as
"This sentence is false."

opinion-based sentences such as
"Chocolate is the most delicious ice cream flavor."

sentences where the truth value depends on the referents:
"I am awake right now." (indexical) "The team went on to win the cup." (context)

sentences with metaphor / poetry / nonsense:
"Anger reflects the clouds."

some counterfactual sentences:
"If the match hadn't ended in a draw, the away team would have won."

See also Are all non self-referential statements true or false?

The OP asks the following:

Can I write or utter any sentence which is neither false nor true?

Yes. An example would be Tomorrow I will rise at precisely 6 am. That sentence today is neither true nor false. However, I will know tomorrow, but by then I will have a different sentence, perhaps: Today I rose at 6:30. That sentence could be viewed as either true or false since it is in the immediate past.

The logic textbook forallx describes three kinds of sentences that grammatically count as sentences although they are neither true nor false: questions, imperatives and exclamations. (pages 4-5)

That text also goes into detail about the kind of sentences to which logicians are interested in assigning truth-values. This is a subset of all possible sentences. These are the ones that logically count: (page 4)

To be perfectly general, we can define an argument as a series of sentences. The sentences at the beginning of the series are premises. The final sentence in the series is the conclusion. If the premises are true and the argument is a good one, then you have a reason to accept the conclusion.

In logic, we are only interested in sentences that can figure as a premise or conclusion of an argument. So we will say that a sentence is something that can be true or false.

What counts in logic are arguments. Arguments are composed of sentences on which one can assign a truth-value.

The OP also asks about self-referential sentences. These are sentences that should logically count. One should be able to assign a truth-value to them and yet when we do we run into paradoxes. These paradoxes are important. They challenge how we deal with that subset of sentences to which we want to assign truth-values. We not only want to assign truth-values. We also want to consistently do so.

P. D. Magnus, Tim Button with additions by J. Robert Loftis remixed and revised by Aaron Thomas-Bolduc, Richard Zach, forallx Calgary Remix: An Introduction to Formal Logic, Fall 2019. http://forallx.openlogicproject.org/forallxyyc.pdf

Is every sentence we write or utter either true or false?

NO. A sentence is "a textual unit consisting of one or more words that are grammatically linked. [... The] words [are] grouped meaningfully to express a statement, question, exclamation, request, command or suggestion.

A question is neither true nor false.

You are asking a very important, and for certain schools of thought, central question!

There has been for a long time and still is the conception that all utterances can be measured in the truth values of true and false. To understand some of the difficulties that arise, let us consider what it means of something to be true or false.

As you already wrote, the only thing that can be true or false is a sentence. Usually, in philosophy there is the conception that a sentence has to mirror a fact in the world in one way or another. This is called the correspondence theory of truth. If the sentence fits the fact in the world we call it true, if it doesn't we call it false.

Historically, there were attempts to order all sentences into these two categories. For this, of course, all sorts of crazy constructs were tested, among them the already mentioned self referential sentences. Bertrand Russell for example had considerable trouble with certain sentences. What is the truth value, he asked, of the sentence:

The current king of france is bald.

One short answer is: this sentence is neither true nor false, because there is nothing in the world that corresponds with its parts. The early Wittgenstein would say a sentence of this kind is senseless or nonsensical.

The same can be said about sentences that are always true (tautologies) or always false (contradictions) which often are the result of self referential structures, but for brevity's sake I will not exemplify these cases.

If we are content here, we can answer your question with no, there are at least three kinds of sentences: true, false, and nonsensical sentences. But let me add that we just looked at propositions, sentences that (claim to) make a statement about the world. Of course, the question could be rephrased, so that those special cases are excluded, and that we only look at propositions. But I argue that we would be missing an important perspective on language and therefore on philosophy if this constraint were applied.

In the Philosophical Investigations the later Wittgenstein claims that there are not only propositions, orders, and questions, but infinitely many kinds of sentences:

23. Wieviele Arten der Sätze gibt es aber? Etwa Behauptung, Frage und Befehl? -- es gibt unzählige solcher Arten: [...]

He continues by listing many ways in which we communicate that deviate from the classical proposition that philosophers used to view as prototypical sentences:

[...] Führe dir die Mannigfaltigkeit der Sprachspiele an diesen Beispielen, und andern, vor Augen: [...]
Theater spielen --
Reigen singen --
Rätsel raten --
Einen Witz machen; erzählen --
[...]
Bitten, Danken, Fluchen, Grüßen, Beten.

These examples he lists include joking, cursing, greeting, praying.
If you follow Wittgensteins thinking here (which I highly recommend to do) we see that not only are there more ways our language works than in making propositions, but that many other philosophical questions can be seen in a different light through the reevaluation of how language and meaning works, but that would be too far astray for this question.

Well, we have to keep in mind that 'truth' is inherently conventional, so that utterances are only true or false within a given set of conventions. For instance, if I were to utter the phrase "I am a philosopher", how we respond to that statement depends on what we mean by 'being a philosopher':

• If 'being a philosopher' means that I identify as a philosopher, then that may change it's truth value at my own personal whim


• If 'being a philosopher' means that I do the activities of a philosopher, then that is dependent on what activities might be considered 'philosophical'.


• If 'being a philosopher' means that others call me a philosopher, then that may change it's truth value at the whim of others.

Adding as an answer because I can't comment yet. Another form of sentence that does not appear to have been mentioned yet is commands. A command such as:

Go away!

is not a declarative sentence and therefore is surely neither T nor F.

I'm not a philosopher, but I disagree with both of your framings of those example sentences.

May I know your name?

in no way by itself implies that I don't know your name. That information is entirely contextual. I may be an audio engineer testing out a microphone. I may be practicing my pronunciation in a foreign language. I think what you are referring to is not truth or falsehood, per se, but the contextual difference between acting deceptively or not.

Also, in your characterization of the statement

This sentence is false

it sounds like you are referring to a similar "lack of any intention to deceive".

So maybe a better form of the question could be, is it possible to neither be honest, nor deceive. I don't know. Maybe a Liar's Paradox in this context could be constructed, where someone sincerely wishes to communicate to someone else that they always lie. If someone is honest but, incorrect does that count? But there are probably other examples.

Again, I'm not a philosopher, but I would bet that you could get into similar Godelian scenarios with honesty/dishonesty that you could with truth/falsehood. There are logic systems that contain things like belief as part of the logical framework, ie. I say X but I believe Y.

Yes you can write a proposition that is neither true nor false. Let us assume that physical determinism is false, and the future is not determined.

Let P be a future proposition (in temporal logic).

If determinism is false, then P is neither true nor false, now.

Example : The electron E will take a value of spin down . , this sentence is probably neither true nor false (in quantum physics).

Another example : This Uranium atom will start its decay process next week.

https://plato.stanford.edu/entries/determinism-causal/

If there is free will, then again : some future propositions are neither true nor false : "John will watch a movie tonight".

This proposition would be neither true nor false if John has free will.

https://plato.stanford.edu/entries/freewill/

If the proposition is self-referential, like : I am lying now , it is impossible for it to be true or false, since its truth results in its falsehood, and vice versa.

These are some of the examples I can think of now, there are others.

https://plato.stanford.edu/entries/logic-temporal/

I'll try to add something to this from the field of semantics within linguistics. I'm not a semantics expert, so if anyone with more experience wants to correct me, please do. (This is also probably why this answer is so painfully syntax/grammaticalization-focused.)

First, I'll attempt to answer the question as posed for well-formed sentences.

There's actually a theory of semantics that tries to carry out this hypothesis, called truth-conditional semantics (this is actually the theory that I learned in class). It's implemented in a far subtler manner than your question frames, however.

In truth-conditional semantics, everything is a function from tuples of entities (people, places, things, ideas, truth values, abstract 137-dimensional entities, whatever) to truth values. Nouns, verbs, adjectives, everything:

• Nouns are functions that map everything they describe to True, and everything else to False. For example, "sandwich" is a function that maps every thing you can call a "sandwich" to True, and everything else to False. Two pieces of bread with mayonnaise between them? If you want to call that a sandwich, the function denoted by "sandwich" maps to True for you. A drawing of a sandwich? If you want to call that a sandwich, that maps to True for you. A pop tart? If you want to call that a sandwich, you'll burn in hell for infinite infinities, but that maps to True for you.


• Verbs that can take a direct or indirect object are the reason why the definition says "tuples of entities" rather than just "entities". For example, "removes" is a verb that maps triples (or pairs) of things to a truth value; "x removes y from z" returns a truth value that is only true when all of x, y, and z fit this situation. Therefore, the triple (a shovel, dirt, the ground) gets mapped to true, but (the ground, a shovel, dirt) does not.

(Wait, aren't nouns functions too?)
(Yes. Functions can be applied to functions. Long story.)

And what about entire sentences? In both syntax and semantics, sentences can be written in a tree structure:

English comes with rules about how to compose each node from its children -- for example, the QP's value is the outcome of the function in the D node applied to the function in the NP node -- and the sentence's (the S node's) value is the outcome of the VP function applied to the QP function.

Since VPs return truth values, this means sentences *are* truth values.

What about questions and commands? A Google Scholar search was unfruitful, so I'm only speculating here. A common theory of syntax says that both are actually normal declarative sentences in disguise, with some transformations to make it into a question or command due to an invisible word influencing the sentence. This isn't really a complete answer; we don't know what that invisible word does with the truth condition.

As a very unsophisticated hypothesis, consider the following:

(1) I wonder if it will rain tomorrow.

(2) I wonder, will it rain tomorrow?

(3) *I wonder it will rain tomorrow.

In (1), "wonder" takes ("I", "if it will rain tomorrow") as its argument tuple. In (2), it takes ("I", "will it rain tomorrow?") as its argument tuple. This may be evidence that "will it rain tomorrow?" and "if it will rain tomorrow" have the same value, and neither of these values are truth conditions. Further supporting that questions and the if-clause in (1) aren't truth conditions, (3), where we plugged in a truth condition, is ungrammatical.

I can't come up with a similar example for commands. But, since the asker was asking about a statement about all sentences, it should be enough to show that some well-formed sentences don't have truth conditions as their values.

Next, I'll address some issues in your rebuttals to why questions are still truth values.

what I just uttered has come out of the truth hood which indicates that I don't know your name, so in this case, the sentence I uttered is a by product of the truth I have

Ooh, juicy, we're flipping straight from the most formal of semantics into pragmatics. The study of conversational implicatures takes care of this -- conversational implicatures are ideas that aren't directly implied by what you say (e.g. "X is a square" must always imply "X is a rectangle"), but rather by what you say if you're obeying certain social rules about conversation. I can totally ask someone "what is your name?" and already know their name; if I do, however, I'm either joking or being weird. As far as linguists studying the meaning of a sentence are concerned, that social consequence doesn't matter in determining the semantics of the sentence itself.

If in your own personal philosophical ventures, however, you would like to count that social consequence as a valid contributor to meaning, I would agree that everything that can ever be said has a truth value attached to it. These social rules about conversation are so well-ingrained in our ideas that we make legal decisions based off of them, and therefore, you can't say a single thing without implying other things that must have truth values about the situation in which you're saying them -- at the very least, the truth that it's appropriate to say the thing you say.

tl;dr- Sentences are modes of communication, so they're not inherently true-or-false themselves. The idea(s) that someone might have upon seeing a sentence may be true, false, both, or neither in various contexts.

Sorry if this sounds super-pedantic, but people get confused about this stuff because they oversimplify things, then they get confused in the cases in which the over-simplifications don't work, and.. well, it's just a big mess. Being a little pedantic's better than being confused.

1: Sentences are about communication, not truth.

Sentences aren't true/false. Rather sentences are an attempt to communicate ideas.

Being pedantic about this 'cause it's necessary to avoid undue confusion.

2: Ideas can be true, false, neither, or both.

Ideas needn't be simply true or false, since they can be neither or both.

Examples:

1. 
   

   True statement:

   
   
   

   
   

   1 + 1 = 2.

   
   

   
   


2. 
   

   False statement:

   
   
   

   
   

   1 + 1 = 3.

   
   

   
   


3. 
   

   Neither true-nor-false:

   
   
   

   
   

   This statement is false.

   
   

   
   


4. 
   

   Can be true-or-false:

   
   
   

   
   

   This statement is true.

   
   

   
   

3: It's possible to be true-or-false.

Regarding Example (4) from above,

4. 
   

   This statement is true.

   
   

we can:

• interpret it as true, in which case its claim to be true is true, and therefore consistent;




• interpret it as false, in which case its claim to be true is false, and therefore consistent.

4: Logical thought is about consistency.

If you're trying to think in a productive way, then your thoughts need to be consistent. So, if you see

4. 
   

   This statement is true.

   
   

, then you're free to interpret it as being true or false, as you like, in one context. You can then regard it as the opposite in another context, so long as the two don't get mixed up.

This is perfectly fine because, so long as you're consistent about your logic, your logic is consistent

Likewise,

3. 
   

   This statement is false.

   
   

is neither true-nor-false because it's not consistent to interpret it as false nor is it consistent to interpret it as true.

Conclusion: Sentences can be interpreted as true, false, neither, or either.

Point being, sentences are interpreted into ideas, which must be consistent, but needn't be context-independent. In particular, it's not always possible to consistently interpret statements as being true-or-false, while other statements can be consistently interpreted as either true-or-false, as desired.

In short, it's all about if a proposition can be consistently interpreted as true or/and false, not about if a proposition is itself true-or-false.

Tangential: Model-dependent realism is based on this.

Scientists typically understand the world in terms of model-dependent realism.

The gist's that we discuss reality in the context of some model, where what's real depends on the model-dependent context.

Examples:

1. Cows can be spherical in some contexts, but obviously not others.




2. "Fictitious" forces can exist in some contexts, but not others.




3. Entropy depends on model, e.g., as in this xkcd.

The point about model-dependent realism here is that what's "true" about "reality" varies by context, including its underlying models. So while the above answer mostly focused on examples with self-referential statements, other claims like

1. 
   

   Cows are spherical.

   
   




2. 
   

   A non-zero gravitational force exists.

   
   




3. 
   

   Entropy is 10 bits.

   
   

can be true, false, neither, or either.

Here's a bunch of zen koans

Take this one: Can the moon be stolen

And convert it to a pair of complementary predicates :

The moon can be stolen

The moon cannot be stolen

Which will you choose?

Try a similar exercise for other koans

In Computer power and human reason Weizenbaum shows an insidious sequence.

• Data – both the word as well as the fact – existed before computers.
  
  (Heck even computers existed before computers! For Turing a computer was a mathematician doing a computation)


• Then when computers happened the management marshalling and corralling of data became an explosion


• In this explosion the pre-computer data (eg handwritten) out of inconvenience became neglected.


• Until – quite like today's deplorables are unpersonned – it was disappeared!
  

Likewise here.

The field of logic especially classical works best with bivalent statements – those for which the "value" True, False can be assigned

• meaningfully


• mutually exclusively


• exhaustively

And little by little it becomes so top-heavy that all other statements become unstatemented!

As an antidote against this tendency I recommend the great buddhist logician Nagarjuna :

Everything is real and is not real,

Both real and not real,

Neither real nor not real.

This is Lord Buddha's teaching.

More light-heartedly my Dad would tell this joke of the math teacher who had to fill in for the literature one
The poem the class was assigned was Tennyson's Charge of the light brigade

Half a league, half a league,

    Half a league onward,

All in the valley of Death

    Rode the six hundred.

And each boy would be stopped by the teacher: Wrong! Next!!

Finally exasperated he shouted:

Don't you know that ½ + ½ + ½ is 1½?

The question is only relevant to assertoric sentences: The sky is blue.

Most assertions about the material world are irredeemably false. For example:

There is a red flower.

We would say this on the basis of our subjective experience of the perception we think we have of what we think of as our material environment.

We could make true assertions by talking not about the material world but about our subjective impressions that we normally mistake for the material world:

I have the impression that there is a red flower.

We don't need assertoric sentences to be true, though. We need to be able to agree about them. People who disagree about the truth of some assertoric sentence cannot cooperate on this basis.

People will be able to agree on the sentence "There is a red flower" if they could all use the sentence "There is a red flower" to describe their own subjective impression of the same material thing, if any. We don't even need to all have the same impression and we couldn't even verify that we do.

And, broadly, this natural process seems to work and we sort of understand why it can work.

Our notion of truth, together with our notion of knowledge, and indeed a range of terms related to them, are essentially meaningful and convenient simplifications. People just talk as if they knew there was a red flower, even if in effect they don't. They do it because they all come to believe from experience that it works more often than not. And when it doesn't work, we can live with a little bruising to our reputation and ego.

The notion of truth is meaningful because we can use it truthfully talking about our own subjective experience. I can say truthfully "I am in pain" whenever I have the impression that I am in pain, because the impression of pain is pain itself. I can talk truthfully of my impression of a red flower whenever I have an impression of a red flower.

We will therefore use our meaningful notion of truth to talk falsely about the material world, but then what matters is that different people mean the same thing when talking of a red flower.

And presumably, by and large, we can do that.

You are talking about the law of the excluded middle, and there are various criticisms of it https://en.wikipedia.org/wiki/Law_of_excluded_middle#Criticisms

It is interesting to contrast the assumption of this law, to the catuskoti of Indian logic https://aeon.co/essays/the-logic-of-buddhist-philosophy-goes-beyond-simple-truth

Ordinary human language is incredibly complex. Logic and mathematics in geberal, involves taking precise defined abstractions, which can have precise defined relationships. Within a certain logical system and set of definitions, it's possible for the law of the excluded middle to hold. But Godel showed in his incompleteness theorems, that at quite an elementary level of complexity, systems of logic have the capacity to say things which seem self-evidently true but cannot be proved true, that these are inevitable.

“Coke is better than Pepsi” and zillions of similar expressions of opinion are neither true nor false. One can define “better” to make it true, to make it false, or to make it continue to be neither.