Trill naming conventions don't work like that. When they're joined, they take the name of the symbiont as their surname. Ergo, Ezri Tigan became Ezri Dax. Jadzia's pre-joining last name was never revealed, but we know for a fact that a Dax symbiont exists because we've seen it multiple times on screen, coming out of or being put into Jadzia or Ezri.
Uh, yes, I know. I'm talking about Mirror Jadzia. Did they ever refer to her as Dax?