In my experience as a voice teacher, I've found that tension in the jaw is often related to tension in the nape of the neck, where the head and neck connect. When I teach people to shake their jaws, more often than not, I find that, if they can find length in the back of their necks, they immediately find greater freedom in the movement of the jaw. To explore this idea further, we'll do the exercise of lifting the head away from jaw, and then bringing the jaw up to the head. It's a challenging exercise, but really worthwhile.
It's important to breathe through all of this, not just doing it while holding your breath.
The first step is to stand aligned, with your feet hip-width apart, you knees not locked, your shoulders wide and with your head floating above your lengthened neck. Now grab onto your jaw with your index fingers and thumbs. Now, instead of dropping your jaw, hinge your head from your ears, and lift your skull away from your jaw. Next, use your fingers to lift your jaw up to meet your skull, bringing your lower teeth up to meet your upper teeth. Then repeat those steps again: head away from jaw, jaw up to meet head. Keep doing that until your your head is as far up as it goes, probably 4 repetitions or so. Once you're up at the top, drop your jaw, i.e. let gravity bring the jaw down, and then, bracing your jaw with your hands, tilt your head to meet your jaw. Then repeat those steps: let gravity drop your jaw, brace the jaw and tilt your head down to meet your jaw, until you get down to the bottom of your neck/jaw range. You should be looking at the floor when your done. Now, head back up to the top, head leading, jaw following; Once at the top, head back down again: jaw dropping, head following. Do each of those several times, and then end in the middle.
Now you can focus on the action of the skull and keeping the jaw relaxed, by holding your dropped jaw and bracing your arms, and then tilting your head to open/close your mouth over and over. Once you get a feel for this, try to do it with no hands, working on the feeling of your breath turning that corner, from your mouth and dropping down into your core.
Next Step: Tongue Flapping, In And Out