I make sure I brush my teeth properly and wash my mouth out with mouthwash BEFORE I start a fast, and after that I leave it alone, I don't put any fluids in my mouth.
I still shower as it is nice, but need to be careful not to set the temp too high as the body can't adjust, avoid baths.
I think if any water did get into the skin I don't think it would be a problem as it will not get to your kidneys/liver , so in fact it is a good thing.
It is just avoiding water that goes into the digestive system that is important as that means your kidneys and liver can rest completely.