I think it's important to not throw out the baby with the bathwater. You are likely allergic to some chemicals
in your full range of personal care products, laundry, or even public exposure (see below). They may or not be fragrances, shampoos, whatever, and they may or may not all be having the same effects.
I have skin issues with certain synthetic detergents - particularly hand soaps, and particularly those in certain public places and businesses. But others have no effect on me. By figuring out which things caused me grief and which ones didn't (took a couple of years), I beat the problem.
I also have eye-watering issues with certain aromachemicals, but I've learned which ones those are, and avoid excessive use. I have also found that rotation of fragrances and auxiliary products like deo and such has prevented skin problems.
I would be scientific about it. You can go to clothes / hair / scented handkerchief / whatever for fragrances, and special products for personal care, and see if it alleviates the problem. Then gradually and carefully reintroduce normal fragrances, shampoos, handsoaps, etc., until you find the culprits. You can use mild products and soaps in the meanwhile. Dreft, Dove, T-Gel & other Neutrogena, etc. work wonders. There is an entire world of hypoallergenic products already out there.
Soaps and shampoos are often the culprit - frequently used too much by perfumistas who want a clean base. "Scrubbers" are dangerous to you in more ways than just odor.