MYSTERY COUGH WONT GO AWAY!?!?

so it started out with a cold about amonth ago. the cold lasted about a week and a half but i was left with a nasty cough and just sick. Soon after 3 weeks with this cough it caused me to throw up cuz it was so bad, couldnt sleep( gets very bad at night). And caused extreme headaches on one side of my head. i didn't know what to do so i went to the doctors. They heard some crackling in the lungs when i breathed in so they said it was settling in my lungs and would begin to be phnumonia. So the put me on the Z pack which lasted 5 days. My cough definelty decreased a bit but now it is still bad and again making me throw up and is worsening at night leaving me with very bad headaches. I don't know what this could be its not normal it wont go away! im only 16 and i know i shouldnt be coughing . also iv been on the Z pack for about a week now and its still not better!