"I know we're taking long shots with a $10-$20K budget, but I'm not afraid of a little cosmetic work - its the mechanics that need to be sound."
The hassle is with a $10K boat the work may be structural, rather than cosmetic.
Choose a boat with GRP (fiberglass) hull, decks and cabin top and avoid Chinese composite (a layer of GRP over plywood).
Solid glass may look like heck, but the structure should be sound, and can be refinished.
Perhaps an old Gulfstar.