I would like to get some answers focusing on what the first mention of "healthy non-aging immortals" is, that grow tired of living.
I went looking for tales of eternal youth. There are a few kinds.
Sometimes the eternal youth is imposed by a lover. That would be the case of Endymion. Or also some variants of the tale of Ganymede.
Sometimes there is eternal youth as a result of an attempt to escape. That would be the case of Daphne (after being converted to a tree).
Depending of what version you get, the myth of the Moon Rabbit could fit in one of the categories above.
Some times the youth comes with a condition. For example, the tale of Kumbakarnan, who – if I am reading correctly – had to sleep for half of each year, and if awoke would die. He was awaken for war, and died in battle.
Another example, which comes closer to what we are looking for, is the condition of not leaving a particular land. Such would be the case of Odysseus and Calypso... or could be the case of Osin and the land of youth. These are a bit close to "grow tired of living", in that these characters left their eternal youth because of a desire to return home.
I am, of course, oversimplifying these myths.
Now, not exactly immortality... however, I think the tale of Yao Bikuni fits the bill. She ate the flesh of Ningyo. The Ningyo are human/fish yokai, eating their flesh grants eternal youth. Said yokai can also cause catastrophes, but those are other tales. Yao Bikuni got married, and saw his husband age and die, many times. She eventaully took her life.
The story tells how a fisherman who lived in Wakasa Province once caught an unusual fish. In all his years fishing, he had never seen anything like it, so he invited his friends over to sample its meat.
One of the guests, however, peeked into the kitchen, noticed that the head of this fish had a human face, and warned the others not to eat it. So when the fisherman finished cooking and offered his guests the ningyo's grilled flesh, they secretly wrapped it in paper and hid it on their persons so that it could be discarded on the way home.
But one man, drunk on sake, forgot to throw the strange fish away. This man had a little daughter, who demanded a present when her father arrived home, and he carelessly gave her the fish. Coming to his senses, the father tried to stop her from eating it, fearing she would be poisoned, but he was too late and she finished it all. But as nothing particularly bad seemed to happen to the girl afterwards, the man did not worry about it for long.
Years passed, and the girl grew up and was married. But after that she did not age any more; she kept the same youthful appearance while her husband grew old and died. After many years of perpetual youth and being widowed again and again, the woman became a nun and wandered through various countries. Finally she returned to her hometown in Wakasa, where she ended her life at an age of 800 years.
According to the book "Death and the Afterlife in Japanese Buddhism", the tale of Yao Bikuni appears widely in the Tokugawa period, that would be 1603–1867, the book also claims there was evidence of a real Buddhist nun Yao Bikuni in the year 1449.
Answer copied over, as requested by bounty owner.