Some things are absolutely essential especially when dealing with exfoliative cheilitis or even in general.
1. Don't lick your lips.
2. Don't bite your lips.
3. Don't forcefully peel off your lips.
4. Get enough sleep,exercise & eat healthy.
It might take a long time for your lips to heal. Regarding the type of moisturizer to be used, you can discuss using coconut oil with your derm & see if it works for you!