Caulk takes only 24 hours to dry properly, as long as the area is around 70 degrees and not super humid. As long as you've prepared the surface well beforehand, meaning all the old caulk is gone, then there shouldn't be any issue with it sticking to the surface. It seems as though water may be leaking behind it, or is already present somewhere behind the area you're caulking. Maybe that's why it keeps getting soft so quickly, because it's never really dry in the first place?
The Dap caulk should be fine, but I agree that silicone is best. I've used the clear stuff in making aquariums and it lasts as long as you'd ever want it to.