I think if it is Cady, then it can definitely work as its own name.
I love Cady and Kate-sounding names. Especially:
CaitlinKatharineKaterinaCatrionaand just
Kate
I don't really like
Katie that much. I think it's a fine NN for a kid but I would probably just go directly to the NN
Kate from day 1. Unless you are talking about Cady - in which case, I fully support.