Welcome to the 'Downs, Blokdog!
Just being in the undying lands does not grant immortality. Even the Valar cannot undo the Gift of Iluvatar. I always got the impression that mortals in Valinor were granted the right to choose the time of their own death, like the kings of Numenor could. There is a quote, which I cannot find right now, that seems to imply that Frodo would have plenty of time to find healing and peace before passing beyond the circles of the world.
Quote:
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in the silmarillion to king tar-atanamir is said:”for it is not the land of manwë that makes its peoples deathless, but the deathless that dwell therein have hallowed the land”
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The answer is in your own post.
Again, welcome to the Barrow Downs!