I'm going to say yes. In order to become a zombie a body must be dead, and generally the soul becomes a ghost after death. Zombies aren't related to the person they were, they're just a corpse that's been animated.
Unless they're the original Romero zombie, which are corpses animated by souls from an overflowing Hell, which might imply they went back to their original body.
I don't believe there is objective official lore on the cosmology of Romero. (Romero seems distinctly uninterested in that sort of approach to storytelling). I know it says so on the poster of Dawn Of The Dead, but that seems more like poetic ominous tone setting rather than a literal lore dump. Everything inside the movie is speculation by characters who don't have answers.
I don't think the above issue really matters too much, though. I'd say that if we accept that souls exist in the Romero zombie universe, then the fact that zombies can retain memories and habits from their lives shows the soul is retained.
I'd say, as a baseline, any setting where zombies can retain some of their personality/memories means that if a soul exists, it is in the body and shouldn't be able to become a ghost.
I can accept a setting where zombies only exist as reanimated husks being puppeted in some way as also having ghosts of those people.
How about a setting where emotions are part of the soul but the body can keep memories?
So a zombie like that could recite stuff and respond, but would otherwise be instinct/reflex driven since there's nothing in it that makes it want stuff, it just looks for food. It wouldn't react to anything you say unless you convince it that it has to listen to get fed
Sure anyone could make any setting. If the creator explicitly make that split, then that's how it is. I just don't think most settings intentionally make such a split.