Any thoughts on damage that can be done by possibly being addicted to cough drops?

See a doctor: A bigger question is why do you need to be taking cough drops all the time. Any cough that has been going on long enough to need to ask the first question should be evaluated by a doctor.

Not many: Cough drops contain menthol and camphor mostly, and I am unaware of any complications of even heavy use of these. Many drops are sugared, and weight gain and tooth decay, gum problems may occur with heavy use. There is no potential for such drops to cause physical addiction, but psychological dependency can occur to many things. Some people are "oral" and need things in their mouths a lot.