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[LOTR] Does destroying the One Ring cure addiction to the One Ring?

Did Frodo or Bilbo continue to crave the One Ring after the ring was destroyed?
gpt
Gandalf says that "Some wounds never heal," and it's implied that he's speaking of the addiction to the Ring and the actual damage that it does to the soul, along with the other wounds that Frodo has from the Morgul-blade at Weathertop and from Shelob's sting.  Bilbo also mentions missing the Ring when he and Frodo reunite at Rivendell, though it's not clear if he's simply old and recalling his memories, or if he's still feeling an addiction.

To me, the clearest passage that shows some addiction comes soon after the hobbits have settled back into the Shire and started rebuilding it.  For context, Frodo had been given a gemstone by Arwen as a gift:

>On the thirteenth of that month Farmer Cotton found Frodo lying on his bed; he was clutching a white gem that hung on a chain about his neck and he seemed half in a dream.  
‘It is gone for ever,’ he said, ‘and now all is dark and empty.’  
But the fit passed, and when Sam got back on the twenty-fifth, Frodo had recovered, and he said nothing about himself.

To me, those are some of the classic signs of addiction - and while those dream states might not be constant, it seemed that he felt the need to wear Arwen's gem around his neck, so he could always have that sensation of something precious near him.

In any case, the Elves definitely acknowledged that permanent damage had been done to the Ringbearers on the deepest level - after all, they allowed Bilbo, Frodo, and even (later on) Samwise to go across the Sea for peace and healing, and Sam had only been a Ringbearer for a couple of hours.