Two questions with "or"

Which of these is correct?

Shall we paint the room red, or do you want to consider more colors?

or

Shall we paint the room red? Or do you want to consider more colors?

I tend to think that both of these are correct, and I would usually prefer the latter. But I'm wondering if there are any reasons why one is better than the other, if one of them is grammatically wrong, or if one of them is objectively preferrable.

The former would be preferred for more serious works of literature. Starting a sentence with or is very informal and not something you should do in an essay or a professional context.

The latter is a good example of something you would say, but not write. That being said, it is also a rule that is commonly broken and therefore accepted to be correct as well.

So, both are good! Just beware of context.

Frankly--and as a pragmatist and not a grammarian, it's six of one, half dozen of the other. Some grammarians might prefer your first example to your second, because the latter begins a sentence with or, when you could just as easily have inserted a comma before or to create a compound sentence.

The first example allows for a for a short pause between the first and second parts of the sentence. The second example allows for a longer pause, perhaps, between the two independent interrogatory sentences.

As I said, six of one . . ..

Grammarians (a.k.a. prescriptivists) love arbitrary rules that they imagine to impose order on language. The reality is that languages change in unpredictable (and to the grammarians, undesired) ways. And the rule about never starting a sentence with a conjunction is just one of those arbitrary rules that have no basis in the ability for people to communicate. Language isn't a logical system; it's a means of communication.

Your question comes down to whether it is better to combine two clauses into one sentence, with a comma between the clauses, or whether each clause should be its own sentence. We should ask, what is the purpose of a period, versus a comma? Is it to indicate a longer pause (imagining that in spoken language, we have pauses, when often we don't)? If there is a long pause in spoken language, can a comma handle it?

With language, there are more questions than answers, and I'll leave it that way.