Have you had an allergy screening?
What exactly are you allergic too?

Also what is wrong with smelling like chlorine?

The alternative body smells are:
1) Body/foot odor
2) Other more complex and bizarre chemicals (i.e. de-odorant, perfume)

Chlorine seems better than these smells.

You mention skin problems.
Swimming and chlorinated water does remove necessary oil from the skin.

Why hurt the skin even more by showering in hot water, and exposing the skin to a broad array of new chemicals (sosp, shampoo, deodorant, moisturizer)?