what you have their are callouses. That's what allows you to play without going in some amount of pain. I don't know how your fingers get stuck though...
I personally had a problem with those callouses, since I play the guitar for 6 hours a day, 5 days a week. They grew, and after a while became so thick that they kind of peeled off, so for a couple of days, I had to play with my bare fingers, and it hurt like hell. So a doctor gave me a medication that's used for hunting dogs' paws, that toughens your skin almost immediately (in a couple of hours actually).
Doing the dishes, wearing skiing gloves, taking a long bath can hurt your callouses. If yours are weird, I guess you could use some very thin sand paper to smooth them out.