If it's truly Rose Rosette, dig the whole rose up and burn it or seal it in plastic and dispose of it in the trash. Don't take chances, especially since you probably won't want to spray with Cygon. Be vigilant of any roses downwind of the infected one, and look upwind for the source and try to eliminate it if you can. Digging up multiflora isn't fun, but it'll help to keep your roses safer.

First, know that I'm not at all familiar with rose rosette.

However, now that I did an online search to see what the damage looks like, I'd like to suggest the possibility of herbicide damage -- namely Roundup or any other product which contains glyphosate -- this because the deformed growth looks very similar.

I have lost three English Roses to Rose Rosette Disease. There is no cure for it at this time. The advice given by HollySprings is right on. Just dig up the infected rose ASAP and dispose of it., it's very contagious. Make sure you get all the roots and be careful you remove the old soil and put new soil before planting a new rose in the same place. The disease is caused by a mite that thrives on multiflora roses and is spread by the wind...