Hello,
What you are having is chronic cough. Some common causes of chronic cough include asthma, allergic rhinitis, sinus problems (for example sinus infection), post nasal drip and esophageal reflux of stomach contents. Infections such as bronchitis or pneumonia can also cause coughing. These infections can be caused by virus, bacteria or fungus.
In order to confirm the diagnosis and rule out any asthma, some investigations need to be done. Transillumination test and X-Ray or CT sinuses, pulmonary function tests, CT chest and upper GI endoscopy should be done to evaluate these causes. Please consult a chest physician and get it done.
It is very difficult to precisely confirm a diagnosis without examination and investigations and the answer is based on the medical information provided. For exact diagnosis, you are requested to consult your doctor. I sincerely hope that helps. Take care and please do keep me posted on how you are doing.
You need to see and asthma specialist - either a pulmonologist or an allergist. Personally, I prefer an allergist because they can also treat the allergy triggers.
I does sound like asthma is a possibility. A chest x-ray is useless in diagnosing asthma. It can show trapped air that may be indicative of asthma, but other conditions result in trapped air as well.
Feel better and God bless.