Unless Rand disappears from the face of the planet, I don't think it's possible for him to hide his existence (or at least the suspicion that he's still alive) from certain people.

• Moiraine and Thom Merrilin are both too clever to fall for Elayne's explanations. If they ever talked about Rand with her, her facial expressions and speech would likely be incongruous with someone mourning the death of her lover/the father of her child. That, combined with the disappearance of Moridin's body (and the fact that the 3 are not concerned about Moridin's disappearance at all) would tell anybody who was looking closely that something was up. As masters of intrigue, Moiraine and Thom would likely puzzle out what happened even if they never got the full truth from Elayne.

• Nynaeve already suspects that something is up, and she too is extremely clever. After Moridin disappeared she would probably be extremely mad and try to have him tracked down. She would always wonder where he went, but might never put his disappearance together with the reactions of the 3 and think Rand is still alive. If she ever saw Rand in his new body though, she would recognize Moridin and try to capture him, which would mean Rand would have to explain himself.

• Perrin is likely to discover that Rand lives no matter what, unless Rand never passes near wolves in his travels. We've already learned that Graendal and Lanfear had recognizable scents to the wolves/Perrin even after they got new bodies. Which means wolves can smell souls. Eventually Perrin would somehow learn that Shadowkiller still walks the land, and could easily track him down to confirm that it was Rand.