Red Raw Dry Patch, Mildly Sensitive, Under Nose In My

red raw dry patch, mildly sensitive, under nose in my mustache above my lip. size of a dime, grows dead gray flaky skin all around it each day. tried Neosporin, aloe, OTC triple antibiotic ointment, plain moisture lotion with vitamin E, and even got prescribed some Muircipoin-ish antibiotic cream. Nothing changes. if it stays moist, the dead skin does not grow much. if I leave it get dry, it can make the gray dead skin circle 3 times per day. no bleeding, not spreading, but it does not leave. I cannot shave, because I work with public and I would have no way to cover it.