After I eat, drink, chew gum etc. I always get an awful, hacking cough, often lasting as long as an hour. This has been going on for about a year.

Definitely abnormal. The common denominator of all the things mentioned resulting in hacking cough is swallowing. During this process you may be aspirating some swallowed contents thus triggering a reflex cough response. Seek help of your doctor, and there are testing methods to evaluate your problem.