It's can be really hard to tell the difference between asthma and being out of breath from the exertion. I guess it comes down to getting to know your body. You are doing the right thing by going to see your doctor as they can prescribe appropriate medication so you can exercise.
Your description of taking a long time to recover and having coughing fits for an hour or so after exercise happens to me. The first time my husband witnessed it he asked if I should quit the team if that was what I had to go through. The fitter I am, the fewer problem I have with asthma and exercise. That being said, I've always struggled with exercise in dry, cold windy weather.
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