In this story, people visit an agency where they can exchange bodies (or, if you prefer, minds) with other people at random. It's a popular form of entertainment, although it is controversial enough to be strongly regulated. If I remember it right, the protagonist is a businessman in his forties or fifties, not happy with the way his life has been going, and something goes wrong while his mind is in a sort of buffer awaiting transfer to a body. An agency representative announces that there has been a dreadful error and everyone's minds in the buffer are now unidentified, but don't worry, we'll get you back eventually. If you find yourself in the wrong body, just pull the cord and it will switch you into another one. This is going to take awhile, and the protagonist keeps finding himself in the wrong body, which always has some residual memories. Once it's a senator who had given a speech about Americans' "God-given right to switch bodies," relating to the controversy over mind transfer. There is certainly going to be a public debate now! Much later, it is clear that something has gone wrong; the protagonist should have found his body by now. He switches into a young man's body, one much better than his own -- no, it has terminal leukemia; this visit to the body-switching parlor was a last treat. He indicates that this is the right one, allowing the young man a chance to live on in his own aging body.
Short story read in an English-language anthology, probably in the 1960s to 70s.