thanks for your question 
i completely understand your question 
cough can be due to the inhaler sometimes the inhaled drug can cause post cough.
but if cough is with mucopurulent , yellow , foul smelling expectoration , it is due to an infection .
you need to 
consult a pulmonologist to rule out other causes like gerd , 
bronchial asthma , bronchitis , 
post nasal drip etc 
pulmonologist will evaluate your cough by 
sputum culture , spirometry and accordingly stage the disease and if neede increase your medications or evaluate if formonide is needed or not.
for now in my opinion u need to take a leukotreine antagonist lime monteleukast and levocetrizine as an anti allergic plus a 
proton pump inhibitors like pantaprazole , this will solve your problem
disscuss these treatment modalities with your pulmonologist so that he can prescribe you , these medications need a prescription
in my opinion you have dry allergic cough
thanks 
feel free to ask more questions 
may god bless you with good health