I have had asthma since I was a few months old. I have found asthma has changed in the kind of symptoms I had through my life.
As a baby, coughing, often at night, was the main symptom. As a child, I had the classic symptoms associated asthma, wheezing, sometimes severe and not controlled by medication available at the time. It could be triggered by numerous allergens, physical activity, stress, infection and so on.
As I got older, the symptoms were not so obvious. Tightness in the chest as if my lungs were in a vice. Doctors would listen to my chest and say I was fine because they couldn't hear the tell tale wheeze, so I had no choice but to wait for the symptoms to worsen, before I was given appropriate treatment.
Things are better now, thanks to peak flow meters and better medication. You are encouraged to manage your symptoms, thereby avoiding that I encountered when I was younger. The last time I had a bad bout of asthma following infection and requiring a stay in hospital, it was heralded by a cough and tightness in the chest. It was not obvious to my family that there was something very wrong. It was my saying I was scared, which was not normal, that alerted them.
I suggest you check with your doctor. He will probably ask you to blow into a peak flow meter and check the result with what is normal for your age, etc. and consider your symptoms. There are other conditions that could cause your symptoms of course. You may well just have an infection that is taking ages to clear, but don't hesitate going to the doctor. You are not making a fuss.