How is it possible to still have a cough from being sick 3 weeks ago. i have taken antibiotic, cough meds, benadryl, (diphenhydramine) its still nagging me. why is this?

Get an x-ray: 3 weeks is serious- call for an appointment and get a chest x-ray.

Entirely possible: Some people may have a lingering cough for weeks after a cold and this for a few may be a sign of asthma which is triggered by infection. About 10% of the colds turn into a bacterial sinus infection which would lead to coughing until the infection is treated. See you doctor to get checked.