Interesting on the Dawn dish soap and I totally understand the reasoning. It is a very gentle soap.
I'd suggest it would depend on the type of skin you have - some shampoos are amazing for skin and hair. RegenePur Dr is what I am using now and it's awesome. Other suggestions would be a vegetable glycerine (pure) cleaner, a little tea tree oil, or Spectro gel. I use spectro on the sensitive parts of the skin of my body and it is excellent although it won't sud up or act like normal soap at all. More like applying congee to your had or something... Strange, and effective.