What makes you think that Gollum lost much of his memory? As for the aging, 500 years in the company of the Ring makes you very lustful of it. Gollum was merely alive after Bilbo took the Ring from him because he had become a kind of wight - fully corporeal, but very close to an evil-serving shadow. He was only motivated by the hope that he'd find his Precious. So I'll agree with you that Gollum stayed alive just because of the Ring. Just as the Mouth of Sauron had probably lived longer than the casual Númenórean because he served Sauron too much & too long, the Ring's lure to the 'Dark Side' elongated Gollum's life.
Also, Bilbo had nothing more to motivate him in life. He abandoned his writings to charge Frodo of them, and he only wrote casually songs, which are shorter & can be done faster. Biblo lived amid happiness, he had found his dream in Rivendell, among Elves. He had nothing to prove anymore, and nothing to take from life anymore. For example, in the Grey Havens chapter, when Frodo & Bilbo meet again, Bilbo seems as relieved that he reached the age of the Old Took. This bet was a small burden, the only thing keeping him in life. The travel to the Blessed Lands was the last thing he could possibly desire. I don't say that Bilbo was tired of life, but that he had seen enough of life, so much that he couldn't ask for more.
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