Rick:
Despite its reputation, chlorine rarely causes problems. Some people develop hives from water. Do you get welst when you take a shower? Swim in the ocean? If there is any way, try to induce the welts and then have a dermatologist see you when they're there (or failing that, take a good picture of them.) I have yet to see a patient who had to stop pool swimming altogether because of sensitivity.
Best.
Dr. Rockoff