I know this thread is old but poor you.
Scalp psoriasis is notoriously difficult to rid yourself of.
I have done it with diet. But even that took six months.
Natural products like diluted tea tree oil may help. Make sure you dilute it, as it is very strong. You only need a few drops literally. Say five drops in your shampoo or conditioner.
Also Manuka honey, coconut oil, aloe vera gel, and hemp seed oil could be soothing. Best to rub into your scalp at night, sleep on an old pillowcase that you don't mind staining with stickiness or oil, and wash out in the morning. Apple Cider Vinegar can help too but can also sting a bit, so don't use on patches that you have picked.
Personally I found the chemicals in most shampoo and conditioner to be terrible for scalp psoriasis. I had much relief when I swapped over to a natural shampoo.
It was more expensive for sure. But it was gentler than those harsh chemicals and artificial perfumes added to standard hair care products.
I also stopped using mouse, gel, hairspray and hot hairdryers.
When I was younger I tried the tar shampoos. They never worked and smelt awful.
Hope you all find relief soon.