If Zombies are dead, why do they need to eat people? They don't digest anything--they're dead! So whatever they eat just stays in their stomachs, rotting. After they ate enough, they would explode--end of Zombie.
Also, if Zombies are dead, then if they killed a person, that person would become a Zombie, yes? So their little bits and pieces would be moving around trying to 'eat' the Zombie who ate them. Which would be the end of that Zombie, right?