Correct me if I'm wrong, but I thought allergic asthma and exercise-induced asthma are two different things?

I have allergic asthma, and exercise actually makes it better, since it suppresses my immune system enough to tamp the allergies down a bit. But I'm sure it's possible to have both kinds.

I would definitely suggest that you do see an allergist, since immunotherapy can actually treat the allergies, rather than just the symptoms.