Ugh. And one thing that happens to me if I bite the side of my cheek is that it happens easily after that. It puffs out or something and then seems to want to settle between my teeth! Been there. It hurts though. I think I would go for a switching behavior. Similar to the strategy used of a hand fidget. If something oral like chewing gun doesn't do it, then what about something in your hands? Do you see a therapist? CBT therapy may help. Working on the root cause of your anxiety is a good thing to ultimately work on as well. I have read about gum chewing but also what about taking a big deep breath when you have the urge to chew? Here's an article on this very thing. https://www.healthline.com/health/cheek-biting But ultimately, you want to get at the root cause for why. Hang in there.
Are you saying you bite your cheeks like someone bites their fingernails, or are you saying you bite your cheeks accidentally? You say you do it when they get itchy -- do you notice when this happens there are already sores on them? Just wondering if this is canker sores, which are often caused by ingredients in synthetic toothpastes, or you're just doing this for no reason at all.