How do you manage dead skin cells on your lips?

It seems like the dry, flaky skin on my lips is becoming more noticeable because of the product, rather than because it is summer.

But then again, using a moisturizing product is a dilemma because the staying power isn't great.

I found it harder than I thought to reapply lipstick every time it wore off.

It used to be fine if I gently wiped my lips with a tissue after washing them sometimes.

I feel like my dead skin cells are becoming more noticeable, maybe because it's not a good way to take care of them haha.

What kind of exfoliation method do you use?

If you happen to use any scrub products, please recommend some too!