Should Pulmonologist Be Consulted For Constant, Thick Cough From The Throat?

I have a cough that starts at the bottom of my throat, it tickles and I begin to cough. No chest pain, no anything below the bottom of the throat but the coughing is hard. Then I have a constant drip in the back, sometimes is very thick but no sinus pain or infection, no fever. I have had this condition for a long time, more than 6 mo, I have had lung x-rays, all clear. been checked for allergies, asthma and nothing. My question is: what kind of doctor should I go and see, lung doctor? ENT? allergy?