Hearing what your vet said is a huge help. Time to go get a harness!
We use a harness for 'quick' outside trips or car rides. We clip her harness into the seat belt to prevent her from climbing into the front seat.
Our girl is no nice to other dogs and we had a real problem walking with her when we first got her. Obedience class introduced us to the prong collar, which looked like a torture contraption to me. I went along with it and we have not looked back. A milli-second pull and she's corrected and not wearing herself out by pulling. She does pull with her harness because she knows it won't get her attention when we correct her. It's really helped us to have more pleasant walks.