Unfortunately, after an infection, it can take up to 8 weeks for a cough to settle down - it's unlucky that you got a 2nd infection just as you were recovering from the first.
At this stage, it's unlikely there's any infection left - so antibiotics aren't going to be helpful. Unfortunately, while cough is an annoying symptom, there's not a lot that really helps with it. Codeine syrups may help reduce the cough a bit, but they don't actually do anything for the lungs, and they have a number of side effects, so are probably not worth persevering with if they aren't helping. The fancy syrups you can get from the drugstore are just plain cough syrup with assorted snake oils.
Unfortunately, you haven't really given enough information for anyone to give accurate advice as to what may be the problem. Personally, I'd follow the first doctors advice that it may need 8 weeks to settle down fully - and then, if it doesn't go back and ask for further tests. But if it really is troubling you, then by all means find another doctor and ask their advice (please note that this does not mean go to the ER for this problem).
Doctors aren't upset if you ask for a 2nd opinion. It's your right to ask. If, for whatever reason, the doctor-patient relationship isn't working out, then it's far better that a 3rd party gets involved before things become too difficult. Many doctors will suggest a 2nd opinion if, they feel that they are unable to reach an agreement with you on a potentially significant matter (from their perspective, if they give up in an argument early, and pass the buck to someone else, they're less likely to get their ass sued if it does turn out that they were wrong).