It SucksSince I was 5 I have had this same problem. It has to be family related; my dad has it, my cousin has it, my sister has had it though it was late in arrival. And now more importantly, my daughter has it. It sucks when I have to suffer with it but to see my daughter in such distress knowing that only time really subdues the burn, itch, pain, is brutal.
We do have a small thing going for us. We live in Alaska and the -40 temps do provide some cooling relief. I routinely step out into the frigid air in my boxers and absorb the cooling (near hypothermic) until my feet and shins go numb. Then I return indoors and quickly get dressed. I find if my mind is occupied the itch does not last long and in fact sometimes it fails to show. Other times such as while visiting Hawaii or while stationed in Louisiana I was absolutely miserable. The humidity plays a huge factor in my experience. My shower routine is: ensure the fan is on, jump in right away without waiting for the water to heat (by heat I mean slightly warmer than lukewarm), wash, dry off and then using a separate towel I dry off again. If it begins I zip outside for the cooling which does not take long and then get dressed.
This may sound dangerous and if you ask my wife she'll tell you it is but driving often helps. The first few miles are particularly challenging.
Somebody mentioned transfer of pain locations or something to the effect. I have tried pricking myself with needles, stiff hair brushes, excessive heat, pulling hairs, scratch pads, sand paper (both rough grit like 60 and the fine stuff like 500) and the cold seems to work the best.
I too have given up on Doctors for this. They have all said the same thing (dry skin, soap irritation) and now it would be awkward to mention it. Besides I don't want any more creams or soaps. One more dry skin diagnosis will only frustrate me; however my child is suffering so perhaps I'll give the docs another try.