“I wanted to be a dancer,” says Fred Astaire, wheezing out a tune on a harmonica with his gangly frame draped casually over a medical couch. “Till I was psychologized.” Astaire plays doctor—a shrink, of all things—in Mark Sandrich’s Carefree (1953), a little-known screwball comedy gem as antic and goofy as Howard Hawks’ Bringing Up Baby (1938) with dance. And what dance! Accompanied by an Irving Berlin score, Astaire and Rogers are at the top of their game in the tale of a therapist (Astaire) who must find the root of the commitment phobia that plagues his new patient (Rogers). […]