I hope I can help you here, please though understand that no one can diagnose or state anything online with 100% certainty. All I can offer is my best educated guess based on what you've written.
And I also think its possible that you may have some post nasal drip. This happens when the nasal cavity makes too much mucous. It can occur for many reasons, but common ones are colds, allergies, some medications, sinus conditions, and a deformity of the nose called a deviated septum. The fact that you are coughing most in the morning supports that; the mucous collects in your throat when you sleep and when you wake up, you cough more to clear it.
Most often, the main symptom of post nasal drip is a cough and a constant need to clear the throat. Here is more information regarding post nasal drip: http://www.webmd.com/allergies/postnasal-drip
A post nasal drip is one of the most common reasons for a chronic cough.
GERD or heartburn is another common reason for a chronic cough, and heartburn happens when there is reflux from the stomach to the esophagus. I can not say that is or is not happening in your case, but it is something to consider and perhaps discuss with your doctors. If your cough was better after a meal, I would suspect GERD. Antacids sold over the counter can help if this is GERD, but do please consult with your doctor before adding any medication, even OTC ones, as they can interact with medications you are taking and your doctor knows your history better than I.
Cough can be a variant of asthma as well, but I note you went to an "asthma doctor" (I assume pulmonologist there) who I hope did lung function testing and other testing and was able to rule this out. Here though is more information on cough variant asthma in case it was not addressed: http://www.webmd.com/asthma/guide/cough-variant-asthma
But your question: I think since you say this cough and tickle bothers you, that you should ask your practitioners about different combinations of medications. If the Claritin is not working, maybe Allegra can; if this an allergy of any type. A visit to an allergist would help determine if these are allergies that you have...though based on what you've written I am not so sure they are. You are, though, telling me this cough intrudes on your life...that makes it worth further investigations. Perhaps a second opinion has merit. Perhaps returning to your original doctors and saying you haven't gotten relief and need more help would be beneficial.
I hope I was able to help you,
Please remember to only rate my answer when you are 100% satisfied. IF you feel the need to click either of the two stars on the left, please STOP and reply to me via the "CONTINUE CONVERSATION" button. Let me know the issue you have and I will be happy to continue further and do everything I can to provide you with the service you seek.
If you give any rating below 3 stars then I will not be compensated in any way for helping you today.