thanks for asking your question
I completely understand your problem
you need to visit a pulmonologist who can get some investigations like spirometry and
sputum culture for pyogenic bacteria if your chest xray is normal.
causes of cough persistent cough can be upper or lower resp tract infections ,
bronchial asthma,
tuberculosis, bronchitis, copd, and some of the cardiac problems.
your pulmonlogist may prescribe you with some broad spectrum antibiotics like cefexime for a week or two, and some inhaled drugs if needed, and may give you some cough syrup like ambroxol which will ease clearing up the sputum and will cause some bronchodilatation.
for
sinus infection u need to take some leukotriene inhibitors like monteleukast or an antihistaminic like
fexofenadine after pulmonologists opinion
thanks /regards
feel free to ask more questions
may god bless you with good health