First is the cough from a problem in side or something you are breathing in? A good test I use is a HEPA filted mask. (the kind you can get in the hardware store with canisters). Wearing that for 5 minutes should clear up the mucus quite a bit if it is something you are breathing in. Easy test and takes only 5 minutes. Try a deep breath with the mask on after 5 minutes and see if you still feel like coughing.
You could also try taking Benadryl and seeing if you can take a deep breath after 25 minutes. But the mask also covers irritants. Both are easy tests.
IF ALLERGY
Claritin probably isn't going to affect the mucus. Very similar but slightly different is Clarinex. Clarinex might slow down mucus if from allergy. You can also try a nebulizer with Sodium Cromolyn soultion. Works in 5 minutes but doesn't last more than a couple of hours. It will slow down the mucus cells that react to allergy.
My allergy is pretty bad from pollens including foods that I eat. But when I go out for exercise I wear a paper mask over my mouth. With a mask I can walk a 12 minute 1/4 mile. With out the mask I walk a 15 minute mile. So you can see how my allergies affect my times.
I'd start with a pulmonologist to clear lung problems or infection. After that might try an allergist.
I also have asthma that only happens when I reach a specific level of exercise and doesn't show otherwise besides allergy induced. Sorry to hear that you are having problems. I now what it feels like. :(
What kind of Dr. are you seeing? Go to a pulmonologist. If you are getting a thick mucous, then there is definitely an irritant there creating it. Most likely they will place you on Prednisone. Not fun stuff, but it takes away the inflamation.