Yes, I really think that had something to do with it. He started writing it down for Bilbo, just as a last task to finish before he died, I believe, but as he wrote it down, and saw where he'd fit in, what his suffering had made possible, I think he would have realized that he deserved the chance to find healing before he died. I can't believe that he only sailed out of despair. I believe he gained hope from telling the story, and realizing that he really had accomplished what he set out to do.