@janoh
Are you doctors you are seeing in the same clinic or facility? They should be working together or at least viewing your symptoms, medical information, and prescribed medications.

When your pulmonogist diagnosed you with COPD did he or she do a pulmonary test?

A cough can be caused by many things including allergies, and post nasal drip (I have this) which is a sign of inflamation in your sinuses. As you saw cough stayed and you saw pulmonogist

I think both specialist are looking at different diagnosis and treating them seperately when they should be address as treating both with best medications possible. Having a pulmonogy test could help determine a more specific diganosis best on test results.

Allergies could be aggravating your COPD if you have it. COPD is, and can, be a serious medical condition. Usually a pulmonogist will order pulmonogy test to determine it.

Your deliema of what you should do about medications signals that you need to talk to both specialists again. I have chronic bronchitis and have the inhalers you mentioned but I also have allergies and my ENT treats them. But they work as a team to see what medications or treatments are what I need specifically to my symptoms and test results.
Good Luck!