How does everyone take care of dead skin cells on their lips?

Hello! I am someone who suffers from dry, flaky lips all year round.

Especially in dry weather like this, my lips get even more chapped and cracked, so I'm really worried.

The biggest problem is that I keep picking at the annoying dead skin cells with my hands or biting them with my lips.
Even though I try to be careful, in the end, the skin always peels off deeply, causing bleeding and a stinging wound.
Even when I apply lipstick or tint, it gets stuck between the dead skin cells and looks messy, and trying to force it out seems to just make my lips more sensitive.

I've been thinking about trying a lip scrub from the store, but I hesitate to reach for it because I'm worried it might be too irritating to my lips, which are already prone to damage.

Do you happen to have your own secret tip for soothing dry, flaky lips at home without irritation or bleeding?

A 'holy grail lip balm' that leaves your lips smooth the next day if you apply a generous amount overnight, or
Recommendations for lip masks are welcome, and if you have a specific routine for softening and removing them while showering, please be sure to share it.
 
I want smooth lips.
I look forward to advice from the experts!