I've seen takes where those bitten become a "lower grade" vampire (i.e. the manhwa Rebirth, where the result is basucally nonsentient) Could a zombie be one variation of that?
Vampire's are immortal and so would not be subject to the disease that causes zombification. The zombie would be cured of their disease and turned into a vampire.
Immortal, yes, but not immune or immorbid. They would definitely become diseased. They would likely become hungry for the flesh of prey and not the organs or blood of its predecessor beings.