I used to get terrible heat rashes on my inner thighes, but I haven't had them this year...
Buy a pair of running shorts- the tight thigh-length spandex-y "bicycle" shorts (but without the seat padding)- for when you exercise. I never exercise without a pair of them on under my shorts, and even when walking/jogging in full sun they are comfortable to wear (i.e. they don't add too much heat).
When not exercising, consider buying some light "bloomer-type" undies- again the kind that go all the way down to almost the knee. Whatever is needed to avoid skin-on-skin contact during the heat of the day. You can always switch to something a bit more traditional once the sun goes down.
Hope this helps!