Which bit is wet? If it is the pad underneath then that will dry, might just have been weed on. A bit like a nappy, its whole purpose is to get wet, but the fleece is the stay dry layer between the absorbent bit and the skin. (Going along with the nappy analogy, wool won't work, the properties of wool make it brilliant to use as a stay dry outer layer for nappies as it allows air to circulate but doesn't absorb moisture unless it is compressed. I used wool trousers overnight on my kids for years to keep the bedding dry!)
The weeing underneath it is a pita, I am reconfiguring my set up to try and stop them doing that, I am fed up of them paddling about their own piddle!