I had exactly the same experience. After I contracted Covid, until last year, it usually took me 2 weeks to stop coughing after the other symptoms went away. I think what improved the situation is exercise. I started to sprint walk a lot since last year and so far I haven’t had any long cough since then.

My son also got asthma after Covid. But he also improved when he grew a bit older (now close to six). We still give him the pumps whenever he had a bad cold, though.

Hope you and your family get better. It is very nasty. It is not life threatening but threatens pretty much everything else.