Im not sure about the eating but mine doesnt, and you should definitly f=give him a bath, it'll help the skin come off. and never!!! never never peel the skin with your fingers, it hurts them. Good luck!!
If he is shedding he may be grumpy and probably won't want to be handled. However, you should bathe him frequently when he is shedding. Sometimes I like to put a little bit of Shed-Ease in there to help them out. Even when they are not shedding you should give them baths often.
Sometimes beardies will eat their shed but mine doesn't. I just take it out once I see it.
I agree with the above i bath mine every day during shedding but i never have it so deep that he has to swim has i think this is cruel..... i've never seen mine eating its shed but i wouldn't of thought it would harm your Dragon in anyway.