This is not normal. You should work at finding a doctor who will help you find an effective treatment for you cough.
There could be many possible causes, and several maybe affecting you. You should be checked for allergies, possibly including allergy SKIN testing and allergy shots considered. You should also be evaluated for GERD as it can cause chronic cough.
Sinus infection has probably been ruled out, but what about non-allergic rhinitis?
Keep working at finding an answer and hold your dr accountable to help you. Remember, he is your employee. You pay for his services, therefore he works for you. If he isn't giving your case the attention you think it deserves, find a dr who will.
Hang in there.
I couldn't say it better than bsmsl. Ditto!
Please do consult your ENT specialist as well as an gastroenterologist. You are having problems with heart burns and obesity also, I suppose. Phlegm in the throat might be secondary to GERD a condition called LPR (Laryngo pharyngeal reflux). LPR might be one of the cause for chronic cough if you do not have sinusitis. Also try to do steam inhalation, drink plenty of warm water, avoid coffee/tea and fizzy drinks for few days.