The wot db has this answer regarding Fain in an interview from Robert Jordan:
In The Great Hunt, who wrote the Dark Prophecy on the dungeon wall in Fal Dara? And why, after Ingtar released Padan Fain from the
dungeon, did Fain decide to go to Toman Head? We know he was rebelling
against Ishamael's orders (he was supposed to follow the Myrddraal to
Shayol Ghul) but why did Fain go to Cairhien and then to Toman Head?
A Myrddraal wrote the Dark Prophecy on orders, as a threat. I might
want to use some of the reasons, so the rest on that is RAFO. Fain
(now amalgamated with Mordeth) was seeking his own power base,
something he would try again with Pedron Niall and Toram Riatin. He
wanted enough power to be able to kill Rand, Mat and Perrin, though
most especially Rand, and to protect himself against agents of the
Shadow. Because of Darkfriend reports, the Myrddraal who wrote the
prophecy already knew who the strangers on Toman Head were, or claimed
to be: Artur Hawkwing's armies returned to reclaim the lands stolen
from Hawkwing's heirs. He knew that they collared women who could
channel, which appealed to Fain/Mordeth, since one disliked Aes Sedai
at best and the other purely hated them. The Myrddraal didn't simply
give this up to Fain, you understand. Fain is one of the few people
who could successfully torture information out of one of the Eyeless.
As for why he went to Cairhien first, he knew the location of the
Waygate there (along with several others and how to read the guidings
in the Ways, this last from Mordeth) and preferred to use the Ways
rather than make the longer cross-country journey from Fal Dara to
Toman Head.
(Source: TOR Questions of the Week, February 2005 to July 2005)