Cough for 3 months. Given montelukast for post nasal drip. Cough worse now. Having lots of irritation while I am breathing. What should I do?

Te-see doctor... Possible etiologies for your symptoms include bronchospasm from any cause, gerd, ongoing sinus problems or a combination of these. If you have ever smoked, you need an aggressive work up to discover the etiology of your cough, including possible malignancy, chronic bronchitis, a form of copd. See your doctor as you will need a more extensive evaluation do that the proper treatment can be rendered.
Pulm function test. These are breathing test done as an out patient which determine is you have functional breathing disorders. You should also be checked for acid reflux.