I use "And" at the beginning of sentences.
I think there are two cases, related but distinct: Dialogue, and Exposition.
Dialogue: Yes, people do this. A sentence is a thought. The next sentence is a new thought. But our brains keep working after we have finished a sentence, so sometimes we have an afterthought that is related, and we begin that related thought in speech with "And".
"And another thing, you mind your tongue with your grandma."
Since you are writing in first person, your exposition is in a halfway house between dialogue and exposition; you aim to give the impression the exposition is actually a character speaking. (Unlike 3rd person, where the narrator is often not thought of as a character in the story by readers).
Exposition: You can exploit this quirk of dialogue in prose, not because it is an afterthought, but to indicate that same "beat" of pause or silence between two actions.
David looked into Harry's eyes, with tears in his own. And pulled the trigger.
This connects the actions.
We could say "...into Harry's eyes. Then pulled the trigger." To me "Then" implies the first action is done, and I don't want to imply that, I want the reader to have the impression David is still looking into Harry's eyes when he pulled the trigger.
We could say, "...into Harry's eyes, and pulled the trigger." To me this seems too immediate; I want that beat of a period at the end of the sentence, to extend that eye contact a moment, and make the pulling of the trigger a more deliberate act.
We could say, "...into Harry's eyes. Then without breaking eye contact, he pulled the trigger." To me that is too flabby to have to explain this, and ruins the moment.
Beginning with "And" does the job I want, so the heck with the rules of grammar. We are writing to entertain, that takes priority over formality.
The (dramatic pause) is what is created; but also the word "And" connects "He sat." to your action; i.e. your action is in response to him sitting, and perhaps saying nothing (the impression I get from your sentence).
, and I,. The coherence of the sentence is not dependent on dependent clauses like this. The third example reduces toHe sat nervous of what he was about to say.... At very least it would need another comma to make it work, ie:He sat, and I, nervous about what he was about to say, [carried on, etc...]. – J... Aug 08 '18 at 15:54, and I,cannot be an aside here, or else the sentence as a whole (which must continue along the lines of "... to say, stood and waited.") makes no sense. That is - the aside is, nervous of what he was about to say,, and the comma after "sat" simply serves to emphasise the separation of the clauses. – psmears Aug 08 '18 at 16:07...about to say, otherwise, and I,stands alone as a type of nonsensical interjection. In this formulation, however, the leading comma aftersatdoesn't really have the superior weight OP is looking for. For that, it's a perfectly valid construction to use a full stop as in the second example. – J... Aug 08 '18 at 16:16