I'm a beginner, so please bear with me. I need some advice.

I just finished installing my first builtin shelf, into my TV nook. It
is a fairly large hardwood shelf, supported on three sides, with a few
trims.

The problem is that one of the supporting walls is not straight. It is
a little bow shaped, and the shelf does not fully meet the wall. That
side is about 2' and I get a bow-shaped gap in the middle of that side,
which gets to be up to 1/4" at the center (I think? working from
memory.)

I am using some corner trim all around the shelf to make things look
nicer, and I was hoping that I could force the trim to meet the wall,
but alas, it turns out there is no support at that portion of the wall
to keep the trim bent that way (no wall beam, just sheetrock).

So, I'm thinking caulk may help (it's next on my list of to-do things).
Questions:

1. What type of caulk should I use, specifically for this problem? This
is not a very accessible shelf, but I expect that part of the shelf to
have a little play over time, so I'd like something that would not
crack too easily. Of course, this would also need to take paint. I was
planning to use regular painter's caulk, before I had this issue.

2. Should I fill in the gap before caulking? If so, with what?

Thanks for any advice.
db