My guess would be the Sith Warrior keeps his or her real name upon being bestowed the title of Darth. For example, Athleena Mawiven becomes Darth Athleena.
As I stated above, the Sith Warrior is never granted the rank of Darth. Just like Scourge before him/her, the Warrior remains Lord. A title is given to the character for display, but that's not part of the story so much as a way to head off complaints that Inquisitors became Darths and Warriors didn't.