One thing that really helps is to do a saline wash. This is unpleasant according to my son but it helps with the healing process. Inflammation can cause sores. Allergies or irritants are often involved too. How we handle our nose can play a role as well. Itching it subconsciously (as my son does with his allergies) or any type of trauma even blowing too hard keeps those sores aggravated and open. Do you sleep with a humidifier? Dry air also aggravates it. And this is crazy sounding, but do you know you can have cold sores and herpes infection in your nose? When the doctor diagnosed infection, did he swab it? It would make sense to do so. Docs often visually inspect things but a swab (which doesn't hurt) helps to be sure and can tell you what type of bacteria you are dealing with if it is staph or impetigo or what. That helps them find the best fit antibiotic or antibiotic ointment. Staph is the biggest reason for chronic sores that are caused by infection in the nose and they make oral antibiotics you take in pill form or an ointment (used by my son) you put on topically. Anyway, that is a long way of saying it could be a couple of things going on or needing further treatment for infection.
So, ways to make it heal are using a humidifier always. Use a saline solution in nostrils. and vasoline or lanolin. Since you have tried all that, I'd see if you do have staph and make sure you have the right antibiotic to get rid of it. Then try these above things to heal the sores once and for all. good luck
Did you ever call the dermatologist and let them know the prescribed medication didn't work? It's very important that a doctor know when something didn't work that they prescribed so they can take a second look at the diagnosis. Did the doctor ever tell you what they thought the cause was? Do you remember what type of medications they gave you?