HI, thanks for using healthcare magic
The constant sinus drainage would be causing the cough.
Post nasal drip of mucus is the most common reason for a
chronic cough.
The other common causes of asthma, reflux.
Bronchitis is inflammation of the bronchi which are airways in the lungs. It is possible that this may be present but it is not likely infectious bronchitis because this would not have lasted this long without needed intervention.
If you are allergic to different things in your environment then the continued exposure would unfortunately mean continued symptoms.
Medications that may help include anti histamines/decongestant combination,
topical steroid nasal sprays such as nasonex, flonase, rhinocort.
If there is no improvement then meds such as
ketotifen or singulair , may be needed.
I hope this helps