No, it wasn't like that at all. This was a special choice Elrond and his children had to make, dating all the way back to the end of the First Age, before the Rings were made. The Rings didn't even enter into it.
The Eldar do indeed grow older, even if slowly: the limit of their lives is the life of Arda, which though long beyond the reckoning of Men is not endless, and ages also.
-Morgoth's Ring: "Laws and Customs Among the Eldar"
The Elves weren't precisely immortal: Being bound to Arda (the Earth), they aged at the same rate as it did. This meant they wouldn't "die" until the destruction of Arda. In Middle-Earth,
they could also "fade" as their spirits (fëar) "burned out" their bodies (hroar). There was a chance that spirit could be reborn in a new body, but they could also avoid this by sailing west to Aman.
So the Elves lived for a very long time indeed. They had this "immortality" from the time they awoke at Cuiviénen, long before the Rings were made. For example, Círdan is likely the oldest Elf in Middle-Earth, going back to Cuiviénen before most of the events in the Silmarillion happened. And he saw Elrond and his companions off.
However, "immortality" didn't automatically apply to Elrond, and by extension his children, because they weren't full-blooded Elves. Among Elrond's grandparents and great-grandparents were three mixed marriages, two between Elves and Men, and one between an elf and a Maia.
The Valar indeed may not withdraw the gift of death which comes to men from Ilúvatar but in the manner of the half-Elven, Ilúvatar gave them the judgment; and they judged that to the sons of Eärendil should be given the choice of their own destiny.
-Akallabêth
At the end of the First Age, the "Half-Elven" (just Elrond and his brother Elros at this point) were told by the Valar to choose which kindred to belong to. Elros chose Mankind and became the first king of Númenor, but having so chosen, he eventually died of old age. Elrond chose Elven-kind. So he was still around six thousand years later at the end of the Third Age.
But to the children of Elrond a choice was also appointed: to pass with him from the circles of the world or if they remained to become mortal and die in Middle-Earth.
-Return of the King, Appendix A, "The Númenorean Kings"
But Elrond's decision didn't extend to his children. Each of them had the same choice to make. They were allowed to delay it for a while, but not forever: The Valar set Elrond's departure from Middle-Earth as the deadline.
An thus it was that Arwen first beheld him again after their long parting; and as he came walking towards her under the Trees of Caras Galadon Laden with flowers of gold, her choice was made and her Doom appointed.
-Return of the King, Appendix A, "A Part of the Tale of Aragorn and Arwen"
When Arwen fell in love with Aragorn, she made her choice to become part of Mankind.
She started aging after that.
However:
With the downfall of 'Power' their little efforts at preserving the past fell to bits. There was nothing more in Middle-earth for them, but weariness. So Elrond and Galadriel depart.
-The Letters of J.R.R. Tolkien, #181
So it could be said that since the destruction of the Rings is what made Elrond leave Middle-Earth, it forced his children's choices. But Arwen had already made her choice.
By the way, it isn't ever said what choice Arwen's brothers Elladan and Elrohir made. Robert Foster (author of The Guide to Middle-Earth) decided they also chose Mankind, from the fact that they didn't sail West on the same ship as their father. But that's not conclusive to me.
