It sounds like your airways have become quite irritable following your cold. This increased sensitivity is what is causing you to cough and may also be causing your feeling of being tight. If you are wheezing, it may involve your lower airways, as well. Antibiotics will not affect this sensitivity. If it is relatively mild, it will probably improve over the next weeks. If this is really a problem, you may need further therapy to reduce the airway inflammation.
I also think you should see a good ENT (ear, nose & throat doc) to find out what's happening in your larynx (voice box)! I hope you'll do this soon.
Some infections can spread to the "vagus" nerve, and this can cause a new case of gastric reflux. Gastric reflux can cause coughing, extra phlegm, the need to constantly clear the throat, a feeling of a lump in the throat (globus), etc.
A good ENT doc can do a laryngoscopy on you, to see what's happening in your throat & larynx (voice box) area. A high up type of gastric reflux, called LPR (Laryngeal-pharyngeal reflux) may cause any of the following signs, that can be seen, during a laryngoscopy: redness (erythema) of vocal cords or throat area, swelling of vocal cords, and extra throat mucus.
A good Pulmonologist (lung doc) can tell you if your lungs have a problem or not.
If "indoor air pollution" is a problem, this can also irritate your upper airway (larynx area) and lower airways (lungs, etc.).
My husband and I were greatly helped (and diagnosed quickly) at Nat'l Jewish. You can speak with a LUNG LINE nurse, about your situation, by calling 1-800-222-LUNG(5864).
Good luck to you!
Sincerely, Concerned lady