When you grow up in a society where you see constant propaganda, you tend to believe the propaganda. You see it in the real world all the time. A maniac spewing lies to followers will be believed, despite how ridiculous and outlandish the claims, because enough people buy into it to allow people to overlook the absolute ridiculous idiocy of what is being said.
In my setting, the drow, and pretty much everything living in my expansive Underdark, live in isolated cities where information from the outside world is rare. Those that doubt the official story are either killed, outcast, or adjusted. Lolth, in particular, as the Lady of Deception, uses misinformation and manipulation to keep the followers in line. The drow in her service generally believe they are doing what is necessary to preserve the drow way of life from the surface threats, and that preemptive strikes and attacks are the only thing keeping the inevitable surface invasion at bay.
Even if they don't believe it they won't do anything. The success rate for armed revolts very low.
If somewhat successful heading for the surface probably the more likely option.
That river that flows through Cormanthyr flowing into the Moonsea seems a decent location.