Usually the insides of drawers are unpainted, but if it is rough or unsightly you can paint them. I wouldn't use a stain but a paint. Do it to suit yourself and what you want to store in them. After all they are unseen until you pull out the drawer

It sounds to me like you can glue down what is still there and fill in the rest with wood filler, then sand the whole thing smooth and paint. If you do that you'll want to use a regular wood glue I think, spread VERY thin over the whole piece to be glued, press, wipe up any glue that squeezes out, lay wax paper over it, and weight it down with books, bricks etc. for a few hours till it dries. An alternative, rather than replacing the entire veneer, woudl be to cut off the damaged part and just replace that. You have to get the same thickness of veneer, cut the old veneer in a straight line with a razor blade, clean it off, make sure it is perfectly flat and even, and glue on a new piece. Let it hang over the edge a bit, and after it dries, trim off with a razor blade or a router with a flush cutting bit. Of course, then you have to stain and finish it to match, which can be tricky. Filling and painting might be easier. Good luck restoring your treasure!

You might try picking up a sample of someone's black chalk paint. CeCe Caldwell, Annie Sloan, American Paint Company or someone. Chalk paint adheres to most everything. They all claim everything and I've yet to prove them wrong. Even my fridge. It's outlandishly expensive, so go for a sample. You must put a clear coat on it, or wax, which you would do anyway after refinishing (no quotation marks necessary) your piece.

The wood looks to be birch. I would apply a preconditioner, and then use a gel stain which gives better coverage than the oil base. With gel stain you kinda buff it on, rather than oil base that let soak in.