There are several possible causes from what you are describing. The most likely is that you experienced an infection and this has caused an increased airway inflammation and irritability. This is being manifested as coughing with or without airway
spasmCoronary artery spasm
Croup
Eyelid twitch
Facial tics
Hand or foot spasms
Urge incontinence
Vascular spasm (wheezing and possible shortness of breath). If this is the case, treating the airway inflammation is the best approach to relieving the symptoms. The absence of wheezing does not mean this is not asthma.
Another common cause is that the initial infection has cause an inflammation in your sinuses which is causing secondary coughing and chest symptoms. In some of these cases, there are no obvious sinus symptoms, although the inflammation is present.
I would suggest speaking with your physician to determine the source of the continued symptoms and addressing this cause. It may take a good physical examination, a lung function test and possibly a sinus x-ray. A good examination is the best way to start this process.