I had this issue, but it was before I started playing melodicas. You may be a victim of the condition known as aging, Lowboy. My dental hygienist told me it’s not unusual for some teeth with old fillings and crowns to eventually become more sensitive. She recommended switching to Sensodyne tooth paste, which resolved my problem.
But also, if you’re biting down on the mouthpiece while swinging your melodica to and fro, that would sound like a recipe for sore teeth.