I have/had a beautiful peony, Terpsichore, that began to diminish a couple years ago. This year it was about gone. Today I dug it up, thinking to move it to a better spot. It looks like there is still a good root and a small sprout, but half of the root mass is black and slimy. Can I save it?

I'd certainly try. Cut off anything that is mushy and plant it somewhere drier than it is now. Good luck, I hope it survives!