Thank you lamp post :-) But I really don't think it is eczema, and if it is dermatitis I'm intrigued to understand why it is affecting just soles and palms. Bit odd me thinks.
The only time my palms do that, is if I've been using different cleaning products- doesn't explain the feet though! Can only suggest exfoliate and vaseline or baby oil and see if that helps.
I have had peeling palms and foot soles but it was part of an allergic reaction to penecillin and I think you'd know if it was a similar allergic reaction, I had a rash EVERYWHERE!