Oh, how I can relate! 5 years ago, I went to the doctor for exactly the same reason. And with the same old, tired diagnosis - "intractable asthma and reactive airway disease with bronchitis..."
This had been going on for... years. Only massive doses of prednisone helped, and we knew this could not continue.

First let's address "...I was never able to use Albuterol inhalers because they instantly irritated and aggravated my constant coughing..." Many years ago, before it was available as a generic, I had to switch from albuterol to levalbuterol - I could not tolerate the raw throat or the shakiness. Now that levalbuterol is available as a generic, there is no reason to tolerate albuterol.

So I returned home ill from my winter getaway - another lung x-ray - and a sharp-eyed radiologist said the magic words "evidence of bronchiectasis, nodules and opacities." Off to a CT scan & a pulmonologist - with the diagnosis of Bronchiectasis and probable MAC. Interim treatment - start airway clearance with Acapella & albuterol (he didn't know about the value of saline)Then a sputum culture and a wait - he said it would be 6-8 weeks, but in 2 weeks he called & said "Pseudomonas" - which started 2 months of antibiotics until we hit the one that worked. In the meantime I coughed and coughed.

Then came the diagnosis of MAC, and as soon as the Pseudomonas was gone, I began the Big 3... and coughed and coughed.

I too felt "...sometimes it just seems like my airways are irritated and raw and the cough is just on an automatic cough loop..." Then I found Mayo Connect and realized there was more to airway clearance. I read and read. And watched videos about airway clearance.

I learned to take a good drink of water and start with some controlled breathing, which seemed to calm the cough.
Then without any neb, I would use the Acapella device. At first, I would immediately begin to cough.
Stop, sip water, deep breaths, Acapella again. Repeat - for up to 15 minutes. Then STOP.
It took 2 -3 sessions like this before I could successfully make it through 3 Acapella breaths and one huff cough.
Eventually I was able to make it through 5 or 6 cycles of clearance. A little sputum would come. But I realized I was accomplishing "something" because later in the day, LOTS of junk would come up out of my lungs when I coughed - especially after walking or bending over.
Next, I began using 2 puffs of my levalbuterol inhaler 10-20 minutes before airway clearance, and taking a short brisk walk. Wow - better results!
Finally, I added saline - but only .9% - my doc wouldn't go for anything else - and it really helped.
This took quite a few months to progress, during which I switched to an Aerobika PEP device, began to use the levalbuterol neb once a day and was taking the 3 antibiotics.

My story eventually has a "happy ending" - off antibiotics for over 3 years, still doing airway clearance and no relapse.

I hope this gives you hope, that you can get through this. This entire group is always here to answer questions and offer encouragement!
Sue