You need to add a dragger behavior to at least one of the layers. If you don't give them their own dragger behavior, CH uses the general dragger behavior for both layers. And one behavior only lets you drag one dragger at a time.
I managed to get both to move at the same time but have no idea what made it possible. However, it then caused the inside layer to drag whenever I'd drag either one of my hands anywhere. So I gave up, trashed my puppet, and started again. Unfortunately, now I'm stuck trying to attach both lower jaws together again.
I've had the same problems. When you select a layer and scroll down on the right side, you can click the + symbol to add behaviors. If you give them their own behavior they'll react at the same time. On the right side of the scene window you can see all of your behaviors with red dots and visibility toggles. With the red dots you can decide which behavior you want to record. You could first only record the hand- or general dragger, then only record the jaw draggers, that way they'll never bother each other.
I don't think the ideal solution is multiple draggable handles.
I've got some ideas, but they might not work, so if you're interested in having me spend some time rigging your puppet, please send it to dramirez|at|adobe.com and I'll see what I can do.
First, a hinged nutcracker jaw is coming soon - should be an easier way to do this.
For the Dragger, you get one Dragger behavior added to the top level of your puppet automatically when you first import (along with Face, Eye Gaze, etc). When you click and drag, that will look for the closest dragger handle and move it. If you added more Dragger behaviors to individual layers/groups, it will try to also move those at the same time. If you added more Dragger behaviors at the top level and arm them one at a time, then you would be able to record one and see how it plays out while recording the next one - otherwise all draggers are always in "ready to record" mode and will move back to their default positions if armed (red dot is on).