The information provided is not enough to say whether your earlier cough is also due to sinus infection
or some other reason like allergy or acid reflux
(acid coming up the food pipe).
Your cough is persisting and there is throat irritation
also due to post nasal drip
means your sinus infection is persisting.
It is already a few weeks means your body immunity is not able to overcome the infection. So you will need a course of antibiotic along with nasal decongestants.
However it is important to rule out the presence of chronic infection in sinus by examining you. If it is chronic, then the course of treatment may differ.
So please consult an ENT specialist
Avoid coughing too much to give rest to the parts.
Take plenty of warm water.
Steam inhalation if it gives comfort to you.
Wish you good health.