The reason you cough is that your body is trying to mobilize mucus and break the secretions up -- your virus load. Fever, too, is part of the body's way of trying to kill viruses. They don't tolerate a high temperature.
The continuing sinusitis is the reason for the persistent cough in my patient population. You can get a pattern of coughing that is similar to bronchitis that is a result of the low-grade sinusitis.