The Elizabeth you know disappears along with the other ones you see in the water. From that moment it will be one universe, with one Booker and one Anna. With the final decision all universes disappear and becomes one (post credits scene). Anna will never be sold and renamed Elizabeth; Comstock doesn't exist any more and neither does the Booker who is picked by the Luteces at the beginning of the game.
All this is possible because destroying the siphon and controlling Songbird with the song from the future Elizabeth breaks the loop of the universes.
That's why at the beginning the Luteces know you won't row the boat and they have a tally of 122 heads at the fair. The Booker you're playing is the 123rd and the only one that survives.
They have the same powers of Elizabeth but they can't do anything because the only way to make Booker's death count in order to "reset" the multiverse is for him to decide to not become Comstock or Booker. The third choice he makes is wanting to go to Paris with Elizabeth. And the only way to make that choice is for him to rescue Elizabeth.
Without the events of the game if you bring Booker directly at the end he would become Comstock or would still be a desperate Booker. In order to break the loop the Luteces bring the 123rd Booker in the "game's universe", he creates his own memories for the different reality (message at the beginning of the game and explanation by Rosalind Lutece at the end), he succeeds in the rescue, he really wants to change, he breaks the loop and only after dying under these conditions the universe is restored.