Can post nasal drip or sinusitis cause a cough that produces a lot of mucous?

Yes: The cough is likely due to irritation at the back of the throat and the mucous is likely drainage from the sinuses.

Sinus: Yes, it can produce mucous. The sinuses are small, air-filled spaces inside the cheekbones and forehead. They make some mucus which drains into the nose through small channels. Sinusitis means inflammation of a sinus. Most bouts of sinusitis are caused by an infection. The cheekbone (maxillary) sinuses are the most commonly affected. http://patient.info/health/chronic-sinusitis