Hi Kevin -
My husband has similar problems, though in his case it is seasonal... But I have a couple things which will hopefully benefit you.
First, you're right - your constant scrubbing with a BHA probably sent yor skin out of balance and it has never snapped back.
The previous post suggested a VERY gentle cleanser - I agree. Find the most gentle cleanser you can. That means irritant free - look for things that do not have fragrances, soap, detergents, even color, if possible.
After cleanser - it seems like your skin is hypersensitive to moisturizers, which makes a dry skin problem very diffifcult! Instead, try using a nice toner, that has a good water binding agent in it. This will provide some moisture, without clgging your pores. BUT - this may not be enough if you are over dry. Look for VERY VERY lightweight moisturizers - try them out. Everybody's skin is different - you may have to try a few things before you find one that provides moisture, but does not make you break out.
Running is GREAT. It will help push toxins out of your body and help keep you healthy. When my husband was training for his marathon, he started breaking out - probably because you get hormonal shifts when you begin a physical regimen. Stay with it and it will normalize.
Hope some of this helps!