Is there a skincare routine that helps exfoliate around the mouth?

It’s not severe, but my whole body tends to be dry. Even though I hydrate diligently, my skin feels rough, and the dryness around my mouth causes dead skin cells to flake off, making it difficult for my makeup to apply and cake. I feel like relying solely on removal might make things worse, so I’m sure there must be a fundamentally better solution. If anyone knows of a good method, please share your experience.