I've had significant sinus issues and had intermittent horrid coughing episodes from all the drainage before my CT that looked like MAC (cultures negative). Docs then started being aggressive in trying to get drainage under control. The bad coughing episode improved with nasal sprays-Astepro and steroid initially and later added Ipratropium. I also do sinus rinses with mometazone using Neilmed's Squeezie. I do them 1-2x/ day depending on on season and intermittently I try just saline without the steroid. Rare coughing episodes now.

Btw, first pulmonologist thought bad CT could be due to sinuses, but Ent was skeptical. I did review and Japanese thought it related, "sinobronchial syndrome," as I recall, and had several studies. (Haven't looked lately.)
Good luck. The coughing can be exhausting.