The Blight may also not have been a great influence on this emmigration, as most of the Dunmer in Morrowind (especially the prominent ones) who state their origins identify themselves as being from the mainland, the only exception being Llensi Llaram, who lived near Khuul.
So, what do the characters actually say?
Imperials hate Dark Elves; Dark Elves hate Imperials...our Dark Elves left Morrowind to get away from the Church and State, and they don't like it when Imperials get all holy and patriotic with them
I'm wondering whether Mach-Na, the Argonian who said that, is referring to the Tribunal Temple or the priests of the Nine Divines when she mentions the Church. The Tribunal no longer exist, even if the priests do, so the Dunmer probably wouldn't be trying to get away from them. Even if the fact that the Tribunal are slain or, as Oblivion reveals, missing, and the priests have kept it quiet it seems unlikely that no-one would have guessed what had occured, given Nerevar's reincarnation.
But, with the Tribunal gone, the Imperial priests are also more able to exert control, and this seems to be a better description of what is driving the Dunmer to leave Morrowind.
As far as the state goes, the ban on slavery may be the reason for traders to have left Morrowind (such as Ulen Athram, from Kragenmoor) but not the majority of the Dunmer in Cyrodiil. Helseth's actions do not seem drastic enough to drive off such large numbers - Morrowind has been ruled by people that many of the population were secretly unsure about for thousands of years, after all.
Dark Elves like me left Morrowind to escape the whole priest-ridden Tribunal nightmare
A Dunmer in Cheydinhal, however, blames the Tribunal Temple (or perhaps their fall?) for her leaving, which either suggests that the fall of the Tribunal is, somehow, public knowledge, or that the priests have power they shouldn't have, and are using it.
So, I'll open it up for debate - why are there so many Dunmer in Cyrodiil?