Sounds like some kind of parasite. If you have other mice, you need to treat them all for mites/worms immediately, or you'll lose them too.

You cannot just cull and replace the mice. Because the new mice that you bring into your house will also get the mites, you have to kill them off.

Also, get a water bottle. It's not hard, just go to the pet store, there's got to be at least 10 bottles for you to choose from, that will work in a 10 gallon tank.