Bronchitis. This sounds like your cold moved into your chest and how you have bronchitis. This typically last for about a month but last 6 weeks or longer in about 25% of folks. It is usually caused by a virus so antibiotics won't help much. You should see your doc to make sure you haven't developed pneumonia or one of the more common forms of bronchitis known as asthma (or cough variant asthma).
Cough. Could also be asthma. I agree entirely with dr. Chappell.
Treatable problem. Sometimes, after an upper respiratory infection, the airways become more sensitive than normal and bronchospasm occurs manifested by a dry, hacking cough that can last weeks to months. This can be treated with a bronchodilator inhaler as well as a steroid inhaler. You would use the inhalers regularly until the cough is under control and then use them as needed if the cough returns. Treatable.