Can I be allergic to soft water, or my water softener?
If you are allergic to the ocean then yes. The ocean has thousands of times more salt concentration than soft water.
When you first install a water softener, it will remove the soap scum build up off your skin. Unfortunately that soap scum was a nice dermal absorption barrier for chemicals in the water.
Some people notice itching after installing the water softener because of the increased dermal absorption for chemicals like chlorine and that is what the itchy reaction is from.
You can add a whole house carbon filter to eliminate the itchy skin, our Hague Watermax doesn’t have this problem because it softens and removes the chlorine all in one unit.