Does t1d weaken your immune system enough to cause chronic cough?

Pretty much the title.

I’ve noticed whenever i get sick i get stuck with a dry chronic cough for weeks if not months. I caught a cold a little over a month ago, all symptoms disappeared after like a week, but I’m still dealing with a cough everyday. It’s not severe, but throughout the day I’ll start coughing.

I’m a 32 yo male, relatively healthy for a diabetic - 5’8 150lbs, and my a1c last i checked was around 7.3.

I’m just wondering if it’s common for diabetics to struggle with chronic cough after sickness? I don’t have health insurance at the moment, so I haven’t been able to go to the doctors. I’m worried it might be something else so I’m a bit concerned. This isn’t the first time this has happened. I’d say I’ve had issues with lingering dry cough for the last few years now.