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.
I use Nivea face wash normally and also exfoliate as this seems to help my scalp.
I didn't always wash my scalp before shaving as I kind of thought it pointless if I was shaving every day. However, I now wash my scalp before shaving. Seems to feel better, somehow.
I wash my head every morning with the same body wash I shower with. Dial Power Scrub with orange oil and micro scrubbing beads. That, and one of those bath sponges made of a mesh.
I usually use shampoo that is available in the shower.
I treat my head like my face, use (dermatologist) prescribed face wash on face and head and exfoliate before shaving (every 2-3 days) with headblade shed
I use Shea Moisture Raw Shea Butter Moisture Retention Shampoo,after I exfoliate my head when I'm about to shave.