you have got tonsillitis and posterior glossits which may cause chronic cough
due to trickling down of secretion in the trachea by a process called post nasal drip(PND).
you may try a course of antibiotic like amoxycillin clavulanic acid
625 mg thrice daily and azithromycin
500 mg once daily, take plenty of fluids.
you may have concomitant asthma for which you must go for spirometry and accordingly inhaled bronchodilators can be prescribed.
if it does not work meet your doctor.