In general use, "tending to droop". Connotatively, "droopy" suggests sadness and discouragement, or lassitude: A cat in hot weather is definitively droopy; the faces and ears of basset hounds are another good example. It's a close antonym to "perky". The connotation is the whole story here, really: The verb "to droop" is a somewhat different animal.
Note that all of this is intransitive. In principle, unseen diabolical Entities can hold you by the ankles by and droop you out a window over a busy street, but you would not necessarily then be droopy; in fact, I bet you'd be pretty lively. I always am when it happens to me. The point is that nobody can make you droopy; they can only droop you: Droopiness is a state of being. You may become droopy in the course of being drooped -- especially if it goes on for a long time -- but that's not quite the same thing. In any case, this transitive usage of "droop" is marginal at best and should be avoided by any means available. If it were me, I'd go for "hang", "dangle", "suspend", or any of a number of others.