If it's NOT an infection problem, it could you only need to increase blood flow to the lesions to promote healing.
I don't really want to reveal this and I don't think anyone wants to hear this, but several years ago I had anal fissures (cuts) that wouldn't heal for a year. A colorectal surgeon who examined me told me take a bath an hour a day (i.e., 2x a day, 30 mins. per session) to increase blood flow to the fissures. After getting out of bath I was told to take a fan to cool the anus to room temperature (or a hair dryer on the cool setting). He cautioned not to wipe the area because a tissue may wipe away natural oils that would promote healing. After wards, I was to wipe a special green cream in the area to further cool the area. He said I could also use Vaseline. In 6 days, everything healed.
Following these principles, you could instead place a warm towel over your mouth for an hour a day. When finished, take a cold towel and place it on the area for about 5 to 10 minutes to cool it. Put your face in front of a fan to evaporate the excess moisture (or maybe skip this part, just experiment). Afterwards, wipe on Vaseline.
I've never tried this. I just thought of this last night so not sure it'll work but maybe worth a try. Make sure the towels aren't too hot. It might dry your cuts too much.