All Aes Sedai get their history lesson on the war of power and the forsaken but to most of them, at least until recently it was just that, history. Yes its important to know how the world got to where it was but most Aes Sedai did not expect the last battle to come in their lifetime.

Moiraine & Siuan knew at the very begining of their careers that hte dragon was reborn as baby and the last battle would happen in their lifetime. We know that Moiraine was smart and I am sure that in between chasing down leads as to where Rand was to be found she did her homework on the shadow and the forsaken to get ready for what was comming.

Then she finds Rand and she actually meets two forsaken in the flesh. At that point in time most Aes Sedai though the forsaken were still bound away and the Last Battle would be something people in the future would have to worry about. Moiraine knew differently.

I don't think its surprising that she was able to pick up on little details and draw conclusions that others could not simply because the average person did not know that forsaken were loose. Rand & co did know but they lacked the education to know much about the forsaken other than stories told by mothers to frighten their children.

Put it another way, if I am walking along the street I may not spot some lose change lying on the ground simply because I am not looking for it. On the other hand if I believe there may be some money lying about and I am activly looking for it, there is a much better chance that I would spot it that someone just out for a walk.