When a demon dies, it is said that their soul re-enters the Abyss, where it gets a new body. What I assume happens on the plane they died in (that is not the Abyss), is that it melts into some vile ichor instantly. Given the fact that demonic essence (soul) is now in a new body on the Abyss, does that mean one cannot resurrect the demon into it's old form?
It cannot be resurrected because it's not dead.
Upon death, a demon (MM 51)
... instantly reforms in the Abyss, its mind and essence intact even as its hatred is inflamed.
As you say, after the moment of death, it instantly gets a new body in the Abyss, which contains its soul. Therefore, it is not dead. Resurrection and True Resurrection act on a dead creature, and therefore do not work on a demon.
A comparable scenario is one in which a (material plane) creature has been reincarnated via the Reincarnate spell, which creates a new body for the soul. Resurrection would fail on the creature's old body, because the creature's soul is not free to return--it is alive, in a new body.
Additionally, it's unclear if demons even have "souls" per se. The MM keeps referring to a demon's "essence" instead of its "soul", and the only reference to a demonic soul is in the description of a Manes, which somehow is a soul, or at least a former soul. I'm not sure what the implications of this wording are, but it's worth noting that wordings are very carefully chosen in the rulebooks.