Is there w way to lock the top beak in place so when I rotate it, it doesn't move up?
There's not a built-in way to lock the beak from moving due to the pivot of the beak's bone. You could move the bone's pivot in edit mode but I'm not entirely sure this won't break other things in the rig.
You could also try using the 3D cursor to rotate the bone instead of the bone's origin like this:
