How should I manage dead skin cells around my lips?

Now that the seasons are changing, dead skin cells around my lips are starting to peel.

It seems like the toothpaste getting on my mouth while brushing is drying out the area around my mouth.

My lips also get dry and flaky easily, so I can soothe that with Vaseline or lip balm, but even if I apply a generous amount of cream around my mouth, it only works temporarily.

My mouth area looks messy. Is there an effective way to fix this?

Should I scrub around my mouth as well?