Tiptree! You're thinking of James Tiptree, Jr.'s "Slow Music." At least, I feel 95% sure you are. (The dying planet, by the way, is Earth.)
ETA: Having been asked to elaborate, I'll try to do so. (Sorry -- I'm new here.)
I am confident that the story (novella, really) in question is "Slow Music" because the story contains all the elements the asker describes, including the specific detail that the male character is resigned to leaving the planet, while the female one is not. Also, this deeply elegiac story is "seriously atmospheric and really sad."
At the beginning of "Slow Music," the male character, Jakko, boards an automated sailboat to take him down a long river on a quiet, peaceful, full-of-animals, but apparently-empty-of-humans future Earth. Along the way, to his great surprise, he encounters a girl about his age, with the forever-memorable name of Peachthief.
We learn gradually (because Tiptree is a master of information management) that the sailboat is programmed to take people to the "River," where they will leave the Earth forever. This is a "post-human" Earth; the people of Earth have voluntarily decided to leave, to upload their consciousnesses to the stars. The human yearning for this upload, this escape, is profound, and eventually seems to gather in everyone. Earth is not actually dying, by the way, in the sense that its plant life and animals are flourishing. But human life there is almost gone.
Jakko is on his way to the River to join his family -- who have left while Jakko was away alone adventuring in the abandoned cities -- before they get too far away from Earth. But Peachthief has decided, stubbornly, to stay behind on Earth. She doesn't want to join the River.
'Suddenly she turned on him, her eyes wide open in the dimness like white-ringed jewels. “Look! Once and for all, I’m not going! I’m alive, I’m a human woman. I am going to stay on this Earth and do human things. I’m going to make young ones to carry on the race, even if I have to die here. You can go on out, you—you
... “I’m not making fun,” he said, shocked. “It’s just that I don’t know what to think. Maybe you’re right. I really . . . I really don’t want to go, in one
way, ” he said haltingly. “I love this Earth, too. But it’s all so fast. Let
me . . .”
His voice trailed off.'
Peachthief asks Jakko to impregnate her: she wants to stay on Earth and raise children. He thinks about it, and about whether to go to the River or stay on Earth with her. They continue on down toward the River together. The story goes on from there.
The story is, as said, "atmospheric and very sad." It's full of references to poetry, to Keats and Yeats and Blake. I had forgotten, until re-reading it just now, just how sad the final paragraphs are.
The novella was first published in 1980, three years after Tiptree's pseudonymous cover was blown, and seven years before she killed herself. It was reprinted in her 1981 collection, Out of the Everywhere And Other Extraordinary Visions. I own the story in the vast and indispensable Tiptree collection, Her Smoke Rose Up Forever. I've found it online as a reprint in Lightspeed magazine, Issue 65 (Oct. 2015):