[Arwen Imladris]

I wonder if Frodo would have been so damaged if he had actually tossed the ring instead of claiming it then loosing it.

He still would have been damaged, perhaps not quite so much, but even if he had not actually claimed the ring he would have been very close. There is also the sword wound in his shoulder, the sting from Shelob and all of the other hardships during his journey. Anyother reason why he left ME is because when he got back to hobbiton, he did not try to fit in. He was not recognised as being important in his own country therefor it was much harder for him to heal.