Punctuating grammatically incorrect dialog

This has been bugging me for awhile. So, we know the rule for joining two independent clauses with a coordinating conjunction such as:

I drove to the store, and I picked up some groceries.

And we know the rules for joining when it's NOT two independent clauses:

I drove to the store and picked up some groceries.

But let's say a character in a story is talking; however, the character speaking is dropping the subject of the first clause:

"Drove to the store, and I picked up some groceries."

In the past I've wanted to leave the comma there (as shown) because the subject is still implied even if it isn't directly stated ... but in the back of my mind, I hear a voice nagging, "it's technically not an independent clause anymore, making it a comma splice ... just like your 2nd example. Get rid of the comma." Any thoughts?

And, yes, I know. People don't speak grammatically correct so the quotation marks grant a degree of flexibility, especially in the work of fiction and not, say, an essay. But like I said, it's one of those things that has bugged me over the years.