Book and show spoilers within
Your memory about Moiraine’s disappearance in the books is essentially correct. She and Lanfear fall through the terangreal doorway to the lands of the fox people (whose name I won’t attempt to spell). Moiraine gets rescued several books later, while Lanfear escapes by other means and with a new face/body/name.
Siuan also doesn’t die in the books.
In the post-episode feature, where Rafe and some of the cast members talk about the episode, Rafe explains why they killed Siuan off: essentially, this marked the end of Siuan being a major character, and the practicalities of making a show mean that having an actor as prominent (and expensive) as Sophie Okonedo would be wasted for the remaining, secondary role that Siuan plays in the books.
So, they let Sophie go out with a big scene.
The same reasoning can be made regarding Moiraine / Rosamund Pike.
The core of the story doesn’t need those characters anymore. And with them targeting 8 seasons at 8 episodes per season, the show has to focus on the core of the story, rather than the various side-arcs that caused the series to run for 14 large books.
Personally, I like some of those side arcs…but I think the decision makes some sense in the context of only having 64 episodes to work with.
