Well, I don't know much about animation, but I'm fairly confident that I find his leg movement weird because of the way he moves his legs, not his lack of head stability.
The legs on the new version do look a bit better though. The whole animation makes it look as if he's running faster than before, partly due to the reduced bounce, I suppose. But if he's going to be sprinting at full pace, shouldn't he be stretching his legs more? Asking out of curiosity.