We have much the same problem here. So far, the generic baby wash from walmart, Equate, Suave Daily Clarifying shampoo, or Ivory soap work for us. I would google "handmade soap", find someone near you, and ask them if they'd make you some Castille soap, which is made from Olive oil, with no fragrance. Or, Google unscented handmade soap.
With my son, we have not been able to use scented laundry products, and especially not any dryer sheets for many years, actually all his life. Even touching someone else's clothing with perfume or laundry products made him break out. My other children and myself had reactions -either upper respiratory, or rash to other laundry/fragrances/soaps as well.
Before you attempt soap making, be sure to learn more about it. I made a batch of Lavender soap when he was a baby, and we both got sick from the Lavender scent.
We've (me) dealt with multiple allergies all my life, weird ones to usual ones. It is very possible to be in contact with a substance for years and not have a reaction, then suddenly you're allergic. It's similar to a thermometer, when it hits that level in your life, then boom, allergy, can't have that anymore.
Weird thing, I ate apples all my life. One day I ate one and started bleeding internally. Yeah, weird. The doc said it was ulcerative colitis. Well, allergy testing proved I was severely allergic. My grandmother of 86 never had a case of poison oak all her life, even though working in the yard and garden. Then boom, she broke out in a horrible case. So it proves, you can go for years with something, no problem, then allergic to it. 24 hr allergy meds work great. I hope you find relief, and I know how hard it is. A varied diet helps (with food allergies). In my research I found that what food you crave most is likely one you're allergic to. Also one you eat, or are exposed to most is likely an allergen as well.
I've also found thst when the body is full of yeast growth, and bad bacteria, it is more susceptible to allergies. That may be a route to check as well.
I hope you find relief.