The story I'm looking for is very, very similar to the YA novel mentioned above, but predates it by at least a decade, probably more. Like A., the protagonist awakens in a new body every day (or at least regularly, perhaps with every sleep period). IIRC, unlike A., he self-identifies as male specifically. The impetus for the first switch, in this case, was something medical: I believe the character first transferred when the original body died, or possibly was being abused (but I'm trying to avoid conflating it with Gould's "Jumper"). There is no explanation, either scientific or magical, for the transference. Did David Levithan write an early short story that later became "Every Day"? The similarities are so striking, but I've been looking for this story for years and only discovered the book last week.
Perhaps Greg Egan's story "The Safe Deposit Box".
"The Safe-Deposit Box" is about a man who wakes up in a different body every day, and dreams of having an identity of his own. (source)
Found in the collection Axiomatic. First printed in Asimov's, 1990.
Furthermore, here is a review that explicitly connects the story and the Levithan novel.
Every Day by David Levithan (Knopf, 2012) is a novel whose main character, A, wakes up every morning in the body of a different person. "The Safe-Deposit Box" by Greg Egan (Asimov's SF, 1990) has the same basic premise, but a very different plot and tone.