Should I make another appointment? And do antibiotics continue to be effective in the days following?

Yes, it is common for antibiotics to continue working in the days following treatment. It is also normal for coughing with secretions to persist even after other symptoms have improved. Using a sinus rinse can help alleviate symptoms by clearing out sinus secretions that may be contributing to your cough. Additionally, staying hydrated by drinking plenty of fluids can also help. You should start to notice improvement in your symptoms every few days.