You have to go to the towel hut and exchange used towels for a new one. You don't need to go through the cruise card deal again, just swap it out.
It became necessary because of their large ships. The beach towels were becoming a hassle for the cabin attendants to deal with, plus the thefts or "lost" towels. It simplified the process. All towels are managed through the huts. On the Elation last week - Carnival still puts towels in the cabins but if you want to get a clean one you went up to the pool deck and changed it at their towel hut. Otherwise you'd have to wait until the next time your cabin was serviced which could be a while. And on the Elation, if you didn't have a towel with you and wanted one, you had to sign it out at the hut with your card.