I'd go to the doctor or urgent care if you can to get a diagnosis.
Something else: Try washing at least your running clothes in free&clear detergent and skip the fabric softener. If you buy a new running outfit or sports bra, wash it before you wear it. Then make sure you change out of your sweaty clothes right away after exercising. If I can't change I at least take off my underwear and bra, which is where most of the sweat seems to collect.
My skin is moderately sensitive. If I accidentally use regular detergent on an occasional load of laundry, it doesn't bother me in my regular clothes, but if I go out running I start itching as soon as I get sweaty -- esp. in the areas you mention.