Damp Linens

So, we've had Walter the albino for about a week and a half now and it seems to be working out pretty well. He's pretty adorable and has taken to me well (although he's pretty shy around most other people). He's been getting plenty of attention and is eating well, little discolored poop and runs on his wheel whenever he's up, but the past few days have given rise to a concern.

Overnight, his linings seem to be getting damp somehow. At first we thought it was his water bottle leaking, so we removed it and replaced it with a bowl, but last night it got damp again. We did find that if the bottle wasn't sitting forward that there was a slight chance it would be left open and water would slowly drip out, so I think that's out for good, unless we decide to pick up another one eventually. Other than that, there is the possibility that he'd knocked over his water bowl last night, but this morning it still had some water in it, so I can't imagine that it would make both of the linings damp. His container is roughly 3.5'x2', with two 2'x3' cloth linings overlapping in the middle.

So the general thoughts around here are that the first time the bottle just dripped all night until it was all wet, then the second night he jumped around in the bowl until he was dripping wet and then had a night on the town, dancing around his container until it was all wet, then peed on any dry spots just to make sure he didn't miss anything.

The real question is this; is it dangerous for it to be so damp where he sleeps? I would assume so, and if that's the case, has anybody else had this happen? How can it be remedied (besides washing/drying linings every day)?