Adopted 4 yr old Dacshund F who I was told had hair loss due to bad diet. Is this true?

When I got her hair has pretty much grown back. She does seem to have a bit greasy or sweaty feel to her coat, it is shiny, and there is a faint odor, but not a bad odor. I was told she was 4 yrs old but she seems older. She has seperation anxiety, as when I left her in my bedroom for an hour she chewed my carpet up by the doorway. Can the nervousness cause the hair loss also? Could it be an immune or thyroid disorder? Her diet is good. She eats Nutro Holistic Adult Small Breed food. She seems to drink a lot more water than usual. She walks for a mile ok with my other 4 yr old male dog. He likes to play, but she seems to not want to play with him. Don't know if its just her personality or if she just doesnt feel well. When at dog park, she runs with the pack with dogs her size, but seems to be frightened when a bigger dog comes around. She is not completely house trained like my male. Take her out and she doesnt go right away, like she doesnt know what going outside means. Shes catching on sometimes though, but still does it in the house occassionaly. This is pretty much her description currently.