Check "cough variant asthma" with your doctor..

I had the same symptoms as yours.

I got bad lingering cough every time after recovering from serious upper respiratory infection. Five years ago, I had terrible cough for three months. I  visited three doctors and tried antibiotics, cough suppressant, and x-rays but all of no help. Finally  a specialist prescribed "asthma inhaler" and it knocked the cough off in two weeks.

I had terrible lingering cough again for a month after recovering from a flu last year.  The same medicine helped me again.