If you think you are wheat sensitive, you might try an elimination diet where you eliminate certain foods then add them back one at a time. I wouldn't want to eliminate gluten if I didn't need to and gluten is more than wheat. You can get wheat-free bread that isn't gluten free.
I wish you luck though because any bread like product I've tasted that is gluten free is pretty gross.
You can't out-exercise poor eating habits.