Why did Frodo feel he had to leave Middle Earth at the of ROTK?

[Pervinca Took]

A question occurred to me today and, having done a quick search, I think this is the best thread in which to ask it. To requote Squatter:

'Alas! there are some wounds that cannot be wholly cured', said Gandalf (III 268) - not in Middle-earth. Frodo was sent or allowed to pass over the Sea to heal him - if that could be done, before he died.

"If that could be done." Much has been said in this thread about the reconnection with the light. The light within Frodo is not mentioned again after Cirith Ungol, where Sam fancies he sees a light glowing, unless it is just a trick of his tears. The evil and horror of the Ring became so great and terrible that it would have extinguished the light, perhaps, just as the Phial of Galadriel had no power within the Sammath Naur itself. Did the light within Frodo revive after the Ring was destroyed, I have often wondered?

However, the *actual* question with which I came to this thread is the title of this post:

Part of Frodo''s sufferings post-quest were the anniversary illnesses. But is to sail West to sail out of Time, or have I misinterpreted the meaning of passing out of Time? (Since I seem to remember a Letter mentioning that the healing of Aman was still "within Time"?) What I mean is: would there even be an October the 6th or a 13th of March in Tol Eressea or Valinor? Would he be protected from these bouts of illness by being outside Time? I know that elvish settlements even in Middle-earth have a timeless aura to them, but they still used calendars, I believe? What is the case beyond the Sea? Are there still seasons and years? * And if not, would this alone prevent the return of illnesses triggered by a date?

* I suppose there must be ... I seem to recall the expression "The Years Of The Trees."