Though it is not very clear from your question, that whether you have known sinus problem or not, but i will try to give you the best answer.
after any allergy, virus or bacterial infection, may take time to clear away completely, specially when there is any focus of chronic infection, like sinuses
, tonsils, ears, teeth. We have to rule out Asthma also in persistent cough. Chronic cough
can also be because of GERD, fungal infection
or as a side effects of some medicine, like ACE Inhibitors for hypertension
Zpak has high resistance to bacteria also. You didn't mention, the result of your throat culture
. it is better to check the culture report, and if it shows any bacteria, then according to sensitivity test, you may take another course of antibiotics. You should also keep in mind that most of the time throat culture report can come negative also, inspite of presence of bacteria.
Role of corticosteroid is increasing these days in upper respiratory infections. it has proven benefit, if taken in proper dosage, on proper indication under doctor's guidance.
So in short, your Doctor have to evaluate you again to rule out other causes of it like Asthma, GERD, etc. , and to consider another antibiotic, steroid, antifungal, or antacid
. You must also do regular gargles, drink plenty of water, and warm fluids, speak in normal voice ( no whispering, singing, shouting, screaming).