Let me make sure I understand what happened.
They were't thrashing before you took them out of the tank, right? You just took them out because the tank was full of diarrhea?
They started thrashing when they went in the tap water and into the fridge? If the tap water was not dechlorinated, that very well could be the cause of the thrashing and maybe the mucous coat shedding.
So, there may be two problems, one that caused the diarrhea and the second being the damage from the chlorine.
Generally, axolotls heal best in clean, cool water. So, without knowing exactly the cause of either problem, I would say keeping them in clean cool water is what you should do now.
Do you have a test kit for your water parameters? You need to check for ammonia, nitrite, nitrate, and pH. Also, what temperature is your tank at? How long have you had the tank? Is it cycled?
Good luck!