A month after sinusitis and bronchitis, my nose and throat are still irritated and I have a dry cough. There is no phlegm.?

Inflammation... After infection, sometimes inflammation persists. This can even result in bronchospasm evidenced by a dry cough. Bronchodilator inhalers may be needed temporarily to control the cough until the inflammation resolves. A steroid inhaler may even be needed. The nasal irritation can be treated with salt water nasal sprays. See your doctor for these treatments.
Give it time. If you are getting better, some times you have to give it time. All people heal differently. If you smoke, then you must stop. If you have allergies, you need to try to avoid the allergins. Remember, you could have picked up something else and it needs to run its course.