The area around my nose is dry except in the summer. How should I take care of it?

Taking care of my nose skin is difficult.

It's fine in the summer, but in other seasons, dead skin cells flake off around my nostrils and on the sides of my nose. It gets worse, especially in the fall and winter when it is dry.

I apply cream morning and evening, but the relief is only temporary, and the dead skin cells come back quickly, making it look unsightly, like dried snot stuck to my nose. When it's severe, the area around my nose and cheeks itches as well. Where I scratched at night, red bumps appear the next day. How can I get rid of this dryness?