If you look up
Dodie on this site, you will see it says it's a diminutive of
Dorothy. So yes, it's legit. However, "My Three Sons" is the only place I've ever seen
Dodie as a nickname for
Dorothy. I remember an actress named Dody Goodman, but I looked it up and her real name was
Dolores.
I agree that
Dodie is terrible and just about any other nickname for
Dorothy is better, even
Dotty.