My hair is on the oily side, so it's kind of different. By dry, do you mean it's flaky? Or just that there's not enough oil itself? Because I get flaky too even though there's a lot of oil. My problem with normal shampoo is I'd get oily roots, but then the ends would dry out and split.
Personally, the flakiness was caused by psoriasis. Could that be what the sores are from? I got rid of that by doing the apple cider vinegar rinse. Another thing the kept it well under control was using pine tar bar soap as a shampoo bar. I ended up making my own, but you can buy it as well.
In your case I'd probably try the Dr. Bronner's stuff first. Strider's skin on his back gets super dry no matter how much fish oil and stuff he eats. He also gets sores from grass allergies once in a while and I use it on him as a dog shampoo as well. I mix:
1/4 cup water
1/4 cup Dr. Bronner's Peppermint Castile soap (the mint will be extremely soothing if you have sores. Strider loves it going on his hot spots and leaves them alone until they heal after a bath)
1 tsp grapeseed oil (NOT olive oil, that stuff doesn't seem to penetrate the hair shaft as well as grapeseed and just makes your hair slimy)
1 tbsp of apple cider vinegar.
When you add the vinegar the soap will "curdle", like go from clearish to cloudy white. When I shampoo with it I scrub it into my scalp really well, and let it sit for the entire shower so the oils can penetrate and the vinegar can destroy all the horrible bacterias. Same thing with Strider, I try to leave it on him for at least 5 minutes. You also don't need conditioner with this stuff.
