Sauron's spirit was indeed not destroyed but became powerless, because he had put most of his power in the One Ring. He had no power left to rebuild himself or a ruling position. He was doomed to roam Arda as a powerless spirit.
As for the Nazgul: they were humans in origin, who were ensnared by Sauron. After Sauron became powerless, they were subject to the Doom of Man: they really died and their spirits went where human spirits go. [Unless they refused to go; I recall from deep in my memory that it is somewhere told that that is possible, but I'm not sure [img]smilies/smile.gif[/img]]
Edit: From Letter 131:
Quote:
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There was another weakness: if the One Ring was actually unmade, annihilated, then its power would be dissolved, Sauron's own being would be diminished to vanishing point, and he would be reduced to a shadow, a mere memory of malicious will.
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<font size=1 color=339966>[ 4:55 AM January 09, 2004: Message edited by: Earendilyon ]