Welcome to the forum. Don't worry the dry skin
below your mouth is due to your sensitive skin. Sensitive skin has the tendency to dry up, get chaffed and it gets cracked giving a rough look. You can You can apply Betamethasone
ointment locally at bed time to get rid of it. Get yourself properly hydrated. Dehydartion, weather changes, low humidity can be a cause of it.
Avoid getting out doors without proper protection on windy or sunny days. Do not stayin hot tubs for a long time. Avoid using harsh, scented body lotions, deodrants, soaps or cleaners.
Check out the toothpaste which you have been using, you can change it and get a milder, herbal one. Apply lip balms, do not lick your lips and take proper diet. Vitamin deficiencies also result in painful peeling of the lips. Take multivitamin
tabs/capsules once daily for a few days. If you have problems, I am there to help you. Take care.