When I moved to India, a common practice here completely changed my hair/scalp’s destiny... oiling. The concept is to oil your scalp and hair in the evening, let it sit overnight, and wash the next day. The oil used will generally be infused with herbs. I started making my own hair oil, which I also enrich with a blend of essential oils, and doing this weekly, and use an ACV rinse after washing. It gave amazing results.
So for your problem, I would suggest to make a hair oil for yourself, using a blend of essential oils good for psioriasis. If you have the patience, you can also infuse your oil with herbs.
I suffer from eczema, and my mom treated my the following EO blend and it helped a lot (it’s good for psiorasis too). And just to back it up, I’d mention that she’s a biochemist and now doing her PhD in EO, so she knows what she’s doing
melaleuca alternifolia 1 ml
lavandula angustifolia 2 ml
helichrysum italicum 2 ml
cymbopogon martinii 2 ml
eucalyptus citriodora 1 ml
melaleuca quiquenervia cineolifera 2 ml
Diluted in 20 ml of St John’s Wort liquid extract
You can use that on all of your psiorasis affected areas.
For your head, I would use coconut oil and add 5% of that blend (5% because the EO are already diluted in the extract, and I aim for around 2% of EO which is good for mild medication). So if you take 200 ml of coconut oil, you will add 10 ml of that mix to it. Apply and massage on your scalp and leave overnight. On the next morning, prepare 1 cup of water with 1 tsp of ACV, and pour it on your head after washing, and don’t rinse.
If you want to infuse the coconut oil, before adding the EO blend, put some herbs in a glass jar, add oil to cover them of an extra inch and leave in the sun for a month, or warm it at very low temperature for 5-6 hours (around 40C is good). I don’t know which herbs you have access to in your part of the world, but neem and vetiver would be good options. I can give the list of what I use for my hair oil but I doubt you can find easily in the US...