I looked at your animation which I liked very much. Sorry, but without knowing how you rigged the eyes, I can't really know what the problem is. If the eyes are parented to the skeleton then the skeleton must me causing the wobbly movement but, I don't know how that would happen unless you animated to joints to do that. By any chance are you using springs to hold the eyes to the skeleton? I usually like to reproduce the problem before I try to fix it but I can't do that without knowing more about your rig. To be honest, I kind of like the eye wobble.