A cough lasting 21 days with normal chest x-ray and TB tests, no fever, usually comes from post-viral airway irritation or a reactive cough that lingers after an infection, especially with your COVID history and skin allergy tendency. This type of cough does not usually respond to antibiotics, which is likely why the tablet and syrup haven't cleared it.
Next Steps
Going ahead with the pulmonary function test your doctor advised makes sense, since it checks for a cough-variant asthma or airway narrowing, which is common and treatable. Warm fluids, steam inhalation, and avoiding dry classroom air can ease the irritation meanwhile.
Health Tips
See your physician again soon if you cough up blood, lose weight, or develop fever or breathlessness.