Can you provide a screen cap of the rigging panel for your puppet?
Okay, without knowing exactly what you want to see the jaw do, my first reaction is to make the head independent, and see how that works. Then, if that isn't helpful and keeping the head independent, try the jaw not independent.
It looks like you added the jaw behavior to the mouth? It might work better, if you copy and paste the lower half of your characters face (in Photoshop) and make it into a layer called Jaw. Then you can add the nutcracker behavior to that new jaw, and play with the flappiness in the properties panel to get it to move how you like.
Hope that helps!