Question Animation Rigging results in unnatural motions in pre-set animations

I set up a simple animation rig to make my character look towards the mouse position (in a top down shooter). However, when you move, if the mouse position is not somewhere in front of the player, there is a very unnatural head-bob, where it looks like the head is bouncing off of the character's shoulders and snapping into the middle of the back.

I've set up some constraints on the axes, but that only seems to make it look better at a standstill; when moving, it still has an uncomfortable motion.

Is there a way I can fix this? Blend constraints don't seem to help

 

I changed the constrained axes to just Y, and the bobbing effect is gone, but it doesn't look as natural now. If there's another way to make it look better, please let me know!