How do I get over my addiction to chewing gum?

Habit not addiction. Chewing gum is more a habit than an addiction. You probably have gotten used to chewing which may relieve stress, or allow you to do something similar to fidgeting. You may use a gradual approach of switching to chewing something that only lasts a few minutes in your mouth, when you have cravings for gum. Find something else that you can do to relieve the momentary stress.