Scattergories does meet all of the criteria. It's not a drawing game, it's a word game. You have to think of things starting with a given letter that match a category. It's a solid party game.
I do actually know of an actual board game that meets all four criteria, but it's a game that one of my professors made so only one copy of it exists to date. It might be available for sale one day and then I'll whore the hell out of it because it's pretty damn good.