[Thin Lizzy]

It sounds like it could be the substrate as your allergies seem to start when you clean the cage. How often are you cleaning the cage?
I clean mine every 5 weeks, I do daily spot checks on the loo area and remove any soiled substrate and clean that area with warm water.
Benadryl is an over the counter medicine.
I know you can get prescription only sprays that are more effective.
There are cardboard face masks you can get and wear when you need to do a cage clean.
I really hope it isn't a serious allergy, but if it is, there is ways of controlling it and still keeping your hammy.