Vintage Hamilton? Can anyone give me some advice here?

Find a reliable watchmaker near you. They should be able to repair the movement for several hundred dollars, if it has just been sitting. They can tell if it’s salvageable. Refinishing the case will likely be much more for replating precious metals if it’s gold. Some might recommend light cleaning of the dial so some patina remains for character.

Looks like a bargain. Google search turns up similar watched for hundreds more. Those watches are in working order and their dials might be in better condition.

Hamilton bought Illinois in 1928 and effectively retired the name. They resurrected it in 1953 as a way of using Swiss movements without diluting the "American made" Hamilton brand.

Your watch looks to be early to mid-50's. A little research should yield a name and it should have some sort of ETA in it. Servicing should be less than a few hundred bucks.

I am confused, however, with your mention of replacing the bezel and case. Why would you?

I agree with @Aurelian, this very likely has a ‘50s ETA and thus should serviceable, likely for $200-$300. I think a cleanup that would come with the servicing would be all you’d need to get a nice looking vintage watch. I wouldn’t replace a the bezel or case.

Epic find and for an excellent price. Just find a decent watchmaker and ask for a service. A good watchmaker will likely clean the dial up a little bit to prevent dust getting in the movement. The case, if it's stainless steel, needs some (really not that much from the look of your picture) attention with some polishing. For the mild scuffs and scratches, I use household brass or silver polish. For deeper scratches, a Dremel or polishing wheel will work better (just take the movement and dial out before using such aggressive techniques to prevent shock and potential accidents.) Your watch may be stainless steel plated, like many of my vintage pieces, in which case, the most aggressive polishing you should do is a tissue/cotton bud/microfibre cloth and the aforementioned brass or silver polish. Lastly, if it were my watch, I'd take off the expansion bracelet and replace it with a cheap leather strap. With that, you'll have an even more killer vintage timepiece. Good luck in getting it running and may you get many good wears out of it!