My dog attacked another dog

My dog attacked another dog today. I came out of the fence and she pushed through, normally she runs out and goes back in almost immediately, but this time she saw another dog down the street and ran towards it. At this point, my second smaller dog got out when I was trying to recall the bigger one. When the smaller one reached my dog and the victims one, the fight broke out, and my dog latched onto the victims dog, near the tail.

We separated them as soon as possible and I got them secured, the owner was obviously very upset, and her dog was clearly hurt, I tried following after to exchange information and offer to cover any and all vet bills, but I was injured breaking them up and lost sight of her.

I’m really not sure what to do. I’m absolutely aware I’m the bad guy in her story, my dog ran down the street and attacked hers seemingly unprovoked. If I see her again, or they arrive at our house, I will still offer to pay the vet bills in full, but I’m unsure how to handle my own dog in the meantime.

I live with my parents and they’ve said that the dogs can no longer be taken out on walks, but I don’t think this will help. We own a muzzle for the bigger dog (due to an attack 2 years ago that was a very different situation and the fault of the other owner as much as mine), but I’m anxious to take her, or even my smaller dog as he is often the instigator, out in public.

Has anyone gone through something similar?