Brief Answer:
inhaled steroids, directly to lungs

Detailed Answer:
HI, thanks for using healthcare magic

As you correctly found, bronchitis can cause a chronic productive cough though if the symptoms have been ongoing for 6 months, it would not be called acute bronchitis.

Acute bronchitis usually lasts up to 3 weeks.

You may have chronic bronchitis or one of the other causes of chronic cough ( a cough that lasts more than 8 weeks). Since your coughing has been going on for 6 mths, you would be said to have a chronic cough.

Inhaled steroids would help best with bronchitis. These would go directly to the lungs to cause relief. Eg budesonide, fluticasone
A prescription from your doctor would be needed.

Since the cough is chronic, other causes of chronic cause would need to be considered eg :(1) post nasal drip- most common cause, undetected by some persons
(2)GERD
(3)asthma
(4)non asthmatic eosinophilic bronchitis- the presence of a type of white blood cell called eosinophils in the bronchi, also treated with steroids. Diagnosed by lab taking a look at the sputum
(5)with a history of smoking, smoking related causes. Would need imaging of the chest

Summary: (1) have chronic cough
(2)acute bronchitis normally lasts a few weeks only but would best be treated with inhaled steroids going directly the chest
(3)would need to consider causes of chronic cough

Congrats on smoking discontinuation

I hope this helps, feel free to ask any other questions

---