I've wondered about that myself but never found an explanation anywhere. My personal theory is that while Almalexia made herself continue to look like a Chimer because she regarded herself as truly divine and was too vain to accept the gray skin/red eyes, Dagoth Ur kept the gray skin as a way of saying to the Dunmer people, "Although I am bound to the Heart and I'm a god, I am still one of you".
Not that such an outlook would have worked for him since the Tribunal branded him as the Sharmat the moment they learned that he had not died (or at least not completely died) back in the First Era and could expose the truth about their divinity.
Earliest references to Dagoth Ur's dunmer coloration would be some of the stories surrounding the Battle at Red Mountain. From Nerevar at Red Mountain:
And then the Tribunal went into Red Mountain and met with Dagoth-Ur. Dagoth-Ur saw what had been done, for his skin had changed as well, and he tried to avenge the death of Nerevar but to no avail. He was driven off and thought dead. The Tribunal found the tools he had been guarding and, through study of Kagrenac's methods, turned themselves into gods.
Thousands of years after their apotheosis, the Tribunal are still the gods of Morrowind and the old ways of worship are remembered only by a few. And the murder of Nerevar is known to fewer. But his queen and generals still fear his return, for the words of Azura linger long and they see the mark of her curse on their people every day.
So according to this story, Dagoth Ur was still alive when Azura cursed the chimer people and turned them into Dunmer. (and got changed as well) In this case, the curse is instituted while Dagoth Ur is waiting for Nerevar's return so that the tools may be dealt with, but after the Tribunal murders Nerevar.
You know, I get a bit of a kick out of this text, since there is a clear implication that Dagoth Ur and Nerevar are responsible for the Dwemer disappearance.
Dagoth-Ur slew Kagrenac and took the tools the Dwemer used to tap the power of the Heart. He went to his dying lord Nerevar and asked him what to do with these tools. And Nerevar summoned Azura again, and she showed them how to use the tools to separate the power of the Heart from the Dwemer people.
And on the fields, the Tribunal and their armies watched as the Dwemer turned into dust all around them as their stolen immortality was taken away.