When I move the camera, or the characters are even slightly to the side, all the pieces of their face move with the camera.
How do I make that stop?

Not sure of your problem .. can u share the project file?


YouTube Tutorials Here

Sure thing.  So if a character is directly in front of the camera, the mouth and eyes and such are all where they're supposed to be (like the first shot).
But if I move the camera, the character's face pieces will move too (as in the second shot).

.

Hi, you can try to switch the camera settings to Orthographic in Project Settings.

Cool, that did it.  Thanks!