I quite liked Ron's idea of carrying a small cloth, too
I end up with very sweaty wrists, sometimes I don't realise and end up sending an Uke flying when they slip off my wrist.. It's at this point that I dry them off.
Like I said, it's not something we do on purpose just to gross you out, and as embarassing as it can be to have to dry your skin before anyone'll touch you - Ultimately, it is natural.