Christopher
Tolkien distinguishes between LR proper (the published version) and the LR process (including drafts written during the time period that Tolkien was creating LOTR).
So I think it is fair to say that in latter, yes
Frodo died, but probably not in
Valinor. It would have been on the lonely isle of the
elves.
Found this on
reddit:
At the time of her lament in
Lórien she believed this to be perennial, as long as Earth endured. Hence she concludes her lament with a wish or prayer that Frodo may as a special grace
be granted a purgatorial (but not penal) sojourn in Eressea, the Solitary Isle in sight of
Aman, though for her the way is closed. - Letter 297, footnote 119
From this, it seems conclusive he stayed with the Elves on the Lonely Isle, not in Valinor, and there was healed and he lived till he died. And also from this:
...but the mythical idea underlying is that for mortals, since their ‘kind’ cannot be changed for ever, this is strictly only a temporary reward: a healing and redress of suffering. -Letter 154
Now consider the meaning. Why does Frodo
need to die? Death was a gift granted to
men (the gift of Illuvatar), and Tolkien likened
hobbits to men in their lineage. Hobbits only lived to be about 100 years in
Middle Earth, and
Hobbit behavior appears closer to simple men than elves.
Why is death a gift that is Frodo's inheritance as a hobbit? Frodo was not bound to
Arda and content with it like the elves. The lonely isle was for his healing, and probably story telling (remember the Cottage of Lost Play), not his immortality. In History of Middle Earth, the early Frodo (Bingo) is especially restless in seeking adventure in lands beyond
the Shire.
Now for the big question:
how did Frodo shape his life beyond the
music of the Ainur that rules the fates in Arda? We know Frodo's 9-10 fingered hands were guided by Providence. The destruction of
Sauron and beginning of the 4th age is a great matter involving beings created by Illuvatar rather than
Ainur, and therefore would be partially beyond the music of the Ainur.
Tolkien describes the creation music as themes. The most interesting answer to your question would explain the musical theme of
Frodo Baggins. Even in this reality called Earth, you can almost hear the music reaching across imaginations.