Great Job I must say!! The need to clear throat post quitting smoking could be due to rebound upper respiratory symptoms. Eat healthy, drink plenty of water, do deep breathing exercise, blow into a balloon to strengthen the lungs, try jogging or brisk walking or cycling to open up the lungs (however pace yourself and do not tire yourself). With time the feeling may go.
Other than this the feeling could also be due to post nasal drip caused by chronic sinusitis due to smoke allergy or it could be due to a bad gastric reflux which is common in smokers and ex-smokers.
Since a confirmed diagnosis cannot be given on net please consult your doctor regarding this. Hope this helps. Take care!
The medical advice given should not be considered a substitute for medical care provided by a doctor who can examine you. The advice may not be completely correct for you as the doctor cannot examine you and does not know your complete medical history. Hence this reply to your post should only be considered as a guiding line and you must consult your doctor at the earliest for your medical problem.