@inquiringminds Three things come to mind – some people do not have asthma, but can have an "overreactive airway" condition that can lead to persistent cough or other symptoms. Not sure if there is a treatment or you just have to learn to tolerate it.
Another is irritation or developing allergy to something in your home. It can be as simple as a new scent in a detergent, a candle or the essential oil places in avaporizer or diffuser, new carpet or a pet. And it may be a newly devloped sensitivity to any of those. For example, I was around my daughter's guinea pigs without incident for tears, sometimes they even used to live with us. After a bad bout of pneumonia, I cannot handle them or be around them for very long.
One more thing to check before you give up – the possible side effects of any other meds you are taking. WebMD and RxList are both places you can look. Another resource is to ask your pharmacist to review all your meds including non-prescription ones, for the cough as a side effect or interaction between neds causing the cough.
I know we would love to believe our docs do this automatically, but I can tell you from experience with 2 very good ones, it doesn't always happen. After 4 months of struggling to breathe 2 years ago, the pharmacist figured out my asthma meds weren't working due to their effectiveness being suppressed by another prescription. A dear friend and his wife were at wits end over his persistent cough a week ago after numerous visits, tests and chest xrays all ruled out illnesses and confirmed that his copd is stable. Finally the doc reviewed his meds, stopped one of them and 4 days later he is fine. I will not name the drug here, but it is a very common one, which many can take without incident, others take for a time then develop issues and some people do not tolerate well.
Good luck with this difficult issue. Let us know what you figure out