https://www.paulmitchell.com/our-products/tea-tree/tea-tree-special/tea-tree-special-shampoo/

Try that for a shampoo, I have greasy fine hair and have used this before and it's helped. It has lavender and tea tree oil so that may or may not irritate your skin. Also washing hair everyday isn't good, it strips the hair of natural oils. So try that shampoo and on the days you don't shower (shower every other day I would suggest) use pure corn starch for your dry shampoo. Sprinkle a little on the roots not too much (you don't want granny hair) and rub it in a little/brush your hair. Corn starch will help suck up the oils so hair doesn't look as greasy. I went to cosmetology school and these were somethings taught to us. If that one shampoo is too irritating, you can get a regular clarifying shampoo.

Something like this one. https://www.amazon.com/Neutrogena-Anti-Residue-Shampoo-fl-oz/dp/B005IHT8KI

For back acne you can try applying a toner to the body like Thayers Witch Hazel. Thayers Witch Hazel is gentle and alcohol free so it will help cleanse your body (back, chest and arms) gently without stripping the skin too much. It will also help moisturize the skin which will help with dryness.

https://www.amazon.com/Thayers-Alcohol-Free-Witch-Formula-Unscented/dp/B001DJDP7C

It's alcohol, paraben, and propylene glycol free so it should work for your dry skin.

Everyone's skin/hair is different these may or may not work for you. Good luck!