In the following learn video, our friend says that you can re-use an Animator, and that you just need to swap out the animations.
How, exactly, do I do that?
Howdy, that was my mistake. You can just duplicate the controller once you have it the way you want it. Then you can put new animations onto the new version of the controller. I was thinking of 3D stuff when I said it would automatically work.
Thanks. Another answer I got was to use Animator Override Controllers. You give it the original Animator as a parameter, and then it gives you a list of animations to override.