> Could I sell an NFT for my birth certificate to raise some cash, while my original birth certificate remains at the state's Vital Records Department under ownership of the state?
Yes, and it's basically equivalent to writing the words "My Birth Certificate" on a piece of paper and selling that piece of paper. It holds no weight whatsoever.
> If so, what exactly would the buyer be buying?
Nothing.
> Just a promise that I won't sell my "birth-certificate NFT" to anyone else?
Not even that, unless you also did that. You can make multiple NFTs from the same thing. Just like nothing would prevent you from writing "My Birth Certificate" on a second piece of paper and selling that too, unless there was some external agreement that you wouldn't.