Cheilitis refers to eczema of the lip. Like all eczema, it can generally be controlled, not cured. On rare occasions, an actual allergy is involved. Perhaps your doctor can consider performing these or referring you for them, in view of the severity of your discomfort. Even if it can;t be cured, however, it should be possible to suppress the inflammation enough with regular treatment to keep the lips from looking embarrassing. Check with your doctor about other modes of treatment.
Best.
Dr. Rockoff