It is so important to find a pulmonologist who is familiar with Bronchiectasis. From what you wrote it sounds like yours may not have much experience with Bronchiectasis. Can you find someone who specializes in your area?

I can’t say for sure if a nebulizer would help you individually, but it is very common for them to be prescribed. Often, albuterol is used which makes the airways dilate so that clearing them is easier. Some use saline also.
Have you had a pulmonary function test? Mine showed asthma and once infections were ruled out ( a long process with bronchoscopy and a variety of cultures that take 6 weeks for results), I was started on Advair. That took care of the little nagging cough I’d had forever. I have little cough now, but my bronchiectasis is mild. Note-my horrid cough had improved after aggressive treatment for sinuses and post nasal drip.

Btw, I’m not on antibiotics and I believe many others are not. It seems to me that most are treated when their sputum is positive, . However people have very different severity and history of infections and treatment varies.
Keep asking questions. It’s a helpful and reasonable
group.