Had bronchitis the past 4 months, on several antibiotics. Can CT show help diagnose better than chest xray?

Not necessarily... Cough can have different etiologies besides acute bacterial bronchitis, which is what antibiotics will help. You could have residual bronchospasm after bronchitis which is best treated with a bronchodilator inhaler and a steroid inhaler. Other diagnoses include gerd, post-nasal drip or a combination of factors. A ct scan will search for alternate diagnoses besides bronchitis.
Bronchitis chronic. Ct has a lot of radiation, you should first do blood tests for fungus, and other less likely infections, if it is not asthma that is causing your bronchitis like cough. Sometimes, if your airways are very tight the docoter may not be able to hear the wheeizing. Make sure you tell the docotr if you have a family history of asthma or hay fever. If you still can't find the answer see lung doctor.