No need to worry about the blankets and brushes. Rain rot is caused by weakened skin from nutritional deficiency and bugs that are in the soil all the time. All horses are exposed to the bugs that cause rain rot all the time. It isn't contagious like that. It can however become infected and need antibiotics.
You WILL see horses in the same living conditions get it and APPEAR to be passing it around but that it not the case. It is because they are often eating the very same things and are all becoming deficient in A at different rates. Some horses need more than others.
I would up the A immediately. Clearly the horse is already sorely deficient.
I use this *given orally over feed not injected * 5cc given once a month for prevention. I give it twice 5 ccs at 7 days apart to treat bad rain rot already in progress like your mare. Then I start on the once a month dose till green grass is up again. It will clear it up very fast. Usually have hair regrowing in 7 to 10 days with no topical treatment. Vitamin A-D - Jeffers