I think there is some confusion here. Adria has been around since D1. She is a vendor in Tristram. You can read about her whole story, but here it is basically.

Adria came to Tristram as a merchant to fight Hell, and the make a bit of money. The three heroes kill Diablo. One of them jams Diablo's soulstone into his forehead in order to try to contain him. This turns him into the dark wanderer. Apparently, this makes Adria super want him. So she pledges allegiance Diablo, and they get it on, and Leah is created. Adria then left Leah with a friend of hers in Caldeum, planning on using her as Diablo's vessel when the time is right. Cain starts trying to find Adria, and learns about Leah. But he doesnt quite know that she was Diablo and Adria's offspring. He figures out that she is looking for info on the black soulstone. Eventually, Adria figures out that Zultun Kulle's head is in the desert. She is captured and tortured by Belial and his cronies. Then she is saved by the gang.

She then tells them that she plans on trapping the prime evils in the black soulstone. She doesn't mention that she planned on turning the soulstone on her daughter, and making her the vessel of Diablo. But not normal Diablo, this is SUPER Diablo, with all 7 of the great evils inside.

With Diablo defeated, Adria ran away. This is where Reaper of Souls starts. She is the only one who knows where Maltheal is at. Eventually the nephalim et al track her down. She spills that Maltheal is in Pandemonium, and then fights you.

And that is the end of Adria and her parts in all 3 Diablo games.