Cold sores are very common with some medications. I use Methotrexate and my mouth has sores on sores. I know how awful these are. However I have not had them in my nose.
There are several things you can do. First, take Folic acid. This over the counter vitamin can be effective to prevent cold sores. For treatment of mouth sores, I use a compounded mouth wash called Mary's Magic Mouthwash. Your dentist can prescribe it or so can your rheumatologist.
If you are interested in talking to your rheumatologist about a prescription medication to suppress cold sores you might ask about leucovorin. It is a high potency folic acid often used in cancer treatment. Since I started using leucovorin I have not had a cold sore. That is over 10 years now.