LOL...I don't know about the losing hair part or the oily skin but the itchies could be the food.
After switching foods it could take 60 days to see results.
Then you will know if that is the problem.
I want to strongly suggest that you reconsider using the RC only because I have been through it with Ellie and ever since she ate it (she was on it for less than a week), we can't get ahead of her allergies. She is always tearing her ear apart now... I just want your furkid to be okay... I hope I'm wrong and it works for you.
Maybe it's worth a try? I would recommend this food for anyone with a liver compromised dog without allergies...
If the K/D is working for you, great but evidently it isn't if she is itching.
The first ingredient (besides water) is corn starch.
Dry L/D has ethoxyquin in it...