How do you deal with dry, chapped skin as well as dry, chapped lips?

Consistent skincare routine. Exfoliate, moisturise, lip balm, weekly clay mask and aloe mask, drink plenty of water.

Drinking water is probably the most important part.

I live in a desert, I just have to keep hydrated. Lotion just makes my skin soft and prone to scratches

For lips: usually lip balm
For skin: body or hand lotion for dry skin

Just rip off the dry skin. what's underneath is usually nice and moist

I usually don't, which is why my lips are usually chapped.

You simply dont. There are always easier options than taking care of your skin

Just hand lotion and chapstick.