When I was in grad school, I took a lab animal surgery class. We had the kit things and sponges too, but it just didn't do it for me like practicing on fresh tissue. It was like, all the sudden, I just got it because fortunately, when we had mice from experiments to cull, me and the other two guys in our research lab would practice a bit on them (both fresh pre-killed mice and anesthetized mice that would be PTS at the end). Nothing complicated, just basic simple interrupted and an occasional horizontal mattress if I was feeling particularly ambitious, but it was good practice. Odd as it sounds, raw chicken might actually work pretty well too. I heard that suggested somewhere. During one of the labs in that class, we practiced ligating the ligaments and blood vessels in a pig's foot, so if your university has a meat lab, they may be able to hook you up. Might even be able to find them in some grocery stores, mmm mmm!
I haven't taken a surgery class yet in vet school, but the vets who taught our surgery class in vet school recommended to the guy in our lab who was left handed to just learn it the right handed way, which he was fine with after some practice.