The bird may kick and fight for up to an hour after, but it is not alive.
This should not be happening in a proper carotid artery cut. The bird should die - as in no reactive movement - within 2-5 minutes. The the op, your cut is either not in the right location or not deep enough to accomplish the task properly. I have a feeling location is the issue since you tried so many times.
I agree with tree hugger that a razor sharp knife can make a huge difference. I used a scalpel with replaceable blades for a while until I learned how to get a good carbon steel knife really really sharp and now I just stromberg's killing knives and keep a diamond sharpening stone (and some good degreaser) close at hand.