I know you said you react to a lot of things but, just in case you didn’t know, most dry shampoos (or eco ones anyway) are based on cornflour. I imagine you could ‘just’ powder your hair and then brush it out. Probably in stages.
Another idea I’ve found works for my dog (but definitely not for everyone) is washing in salt water (and not rinsing the salt out). It’s a variant of the no-poo so you may have already tried it. It leaves a bit of salt in your hair which absorbs oils and can be brushed out over a few days. Might be better for short hair than long though. And your skin needs to tend to being oily - would dry your skin and hair.
And last but not least: I oil my hair instead. Olive oil or argon oil or whatever stable oil you can find that doesn’t smell too much for you. Test it on a bit of skin for a day or so first (in case your skin doesn’t like it). I heard it about it when reading up on romans. They oiled rather than soaping. (But they also bathed). Oil is a good solvent for the oils in your hair. Less is more, and you do need to brush (to clean your scalp) and towel (to clean off excess oil and any of the dirt the oil lifted).
I have no idea if any of these will help but the last one was what I did for a month at a time for years (oil, brush, put in two braids, leave until messy, then brush & rebraid, ...). It means that when you do get to the soapy shower (yes fragrance free dishwash can be much more effective if you can stand it and rinse really well) your hair looks amazing. And in the meantime your braids look much tidier.
These days I alternate.
It’s just stuff I’ve tried.
Everyone’s hair and scalp react differently.
My only advice is what my grandmother told me (from wartime): keep up the brushing, especially of your scalp. If you can. The thing that causes problems is anything that stays on your skin for too long and breeds irritations.
It’s all a lot of work. I’m sorry.
