There must be a different way to solve the problem
There isn’t. As a lifelong allergy sufferer, the only sure way is elimination.
Re: products. In the ingredient list, the higher up any ingredient is, the more there is of it. Ingredients that give me issues are:
-anything that starts with benzo (preservative)
-limonene (fragrance)
-cinnamon or any derivative
-any combination of laur- and sulph- (foaming agent)
Also consider it may be the combination of shampoo, conditioner, body wash PLUS the detergent that is doing you in. Cut one product at a time to see if it makes a difference.
As a last resort, if you live in a hard water area, get a water softener installed. Am currently staying at a friends with lovely neutral water and my skin has magically cleared up.
Re: itchy mouth. Sometimes a perfect storm of allergens will cause your body to start reacting to anything and everything. It may actually be your washing powder in the first instance and fruit irritating it. It may be tree pollen. It may be grass pollen. It may be your toothpaste. It may be a combination of all of the above. In the meantime, the worst offenders tend to be stone fruits (peaches, plums, cherries, all the good summer ones), kiwi, and bananas.
Hope that helps.