Yeah, my car had a couple spots like that when I bought it, but a couple passes with polishing compound fixed them. (Start with Turtle Wax polishing compound, then follow up with the Meuigar's
ScratchX. Use something like an old T-shirt for application.) If you hit it with Nu-Finish afterwards it should shine like new and water will bead up and roll right off!
Sandpaper just makes lots of little scratches, and the compound should level them off and fill them in enough to not be noticeable.
Of course test these products on an inconspicuous spot before you do anything with them so you know they won't screw anything up. (They shouldn't as long as you're not too aggressive.)
The wax may need to be re-done every so often depending on how often you play, but you get more instant results than applying new polyurethane and you don't have to worry about toxic fumes filling the dorm.