does Frodo truely find the peace he deserved in the grey havens?

[Galin]

Part of the source in question might be letter 246 (drafts, dated 1963, to a Mrs Elgar), where Tolkien notes that both Frodo and Bilbo still bore a mark of the One, and:

(...) '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 Sea to heal him -- if that could be done, before he died. (...) So he went both to a purgatory and to a reward, for a while: a period of reflection and peace and a gaining of a truer understanding of his position in littleness and in greatness, spent still in Time amid the natural beauty of 'Arda Unmarred', the earth unspoiled by evil.' (...)

JRRT, Letters

Full context is best. Much later in 1971, JRRT wrote to Roger Lancelyn Green (in part):

'As for Frodo or other mortals, they could only dwell in Aman for a limited time -- whether brief or long. The Valar had neither the power nor the right to confer 'immortality' upon them. Their sojourn was a 'purgatory', but one of peace and healing and they would eventually pass away (die at their own desire and of free will) to destinations of which the Elves knew nothing.'

Anyway, I think letter 246 is maybe part of the source of what you read online.