When the vet gave antibiotics, did the vet actually say there was an infection? Or, where they given as just "let's try this"...? What color is the gunk? If it's greenish...that's almost always an infection.
What else is going on w/ the nose? It's not from moisture. Is it the area up where the nose meets the head - or is it down on the snout? Is she itching that area? Bc the itching can def make it raw. They should do a skin scraping of that area on the nose.
Is she itching anywhere on the body? Chewing or licking her paws? If those symptoms are present, then I'd suggest you see a Vet Derm first - to see about allergies. Vet Derms are simply amazing at getting skin issues addressed properly. We see a vet derm and we LOVElovelove him
! When there, have them peek at the teeth - as someone above mentioned.
The other angle is to see a Vet Ophthalmologist - as they have instruments to see they eye that regular vets do not have. One member here at YT had a kiddo whose eyes had chronic issues - she'd been to her vet several times; she then went to an Ophth and they found this micro-hairs/lashes growing inside her lids, which irritated the eyes all the time.