One more thought…I had horrid coughing episodes for a couple years along with a little nagging cough. Azelastine was a big help for the horrid cough episodes, but didn’t stop it completely. It can be OTC or prescription strength.
I went to Mayo due to a lung issue and they started a compounded drug they mix for all the secretions. It has mometazine, Ipratropium and diphenhydramine. I use 2x daily and I have an Ipratropium spray to use for break thru since it last 6 hrs and other 2 are 12. I take a class 1 antihistamine at bedtime.
The little nagging cough was thought to be from asthma (new diagnosis), and the steroid spray stopped it. These work pretty well for me.
Here’s the Mayo description:
“Cough Clinic specialists have pioneered a triple combination nasal spray that uses antihistamine, anticholinergic and steroid medications. Mayo Clinic pharmacists dispense the spray as a compounded formulation.” A combination of sprays in these categories would work so your doc wouldn’t have to order Mayo’s, but is convenient as 1.

I hope you find a solution. That coughing is exhausting.