When Gracie Anderson sings “Come On-A My House” in “Tenderly: The Rosemary Clooney Musical,” it does exactly what it’s intended to do: Welcome you into a nostalgic good time at Old Log Theatre.

I’m not sure Anderson sounds much like Clooney. But Anderson has warmth and a big, honeyed voice, and she commands the stage the way a concert soloist like Clooney would have to. Anderson inserts improv-y little bits into her dialogue and slides in and out of vocal impersonations, all of which helps get across that Clooney was a small-town gal with a distinctive, big-time talent. She had the smarts and personality to match wits with Frank Sinatra, Bing Crosby and other stars of her day.

“Tenderly” is not as assured, but it begins boldly, with an onstage breakdown — the real one Clooney had shortly after her friend Bobby Kennedy was killed in 1968. The action shifts to a mid-century-modern psychiatrist’s office. There, Clooney and her therapist (versatile C. Ryan Shipley) work through her depression and her addiction to pills, constantly breaking out of the office to enact scenes from Clooney’s life, with Shipley assuming the roles of everyone in it, including Sinatra and Crosby (in the same song!), as well as two-time husband José Ferrer and Clooney’s mother.

The conceit works as a way to power the audience through what is essentially a jukebox musical, filled with Clooney’s hits. But the herky-jerky structure keeps the show off-balance. Is it a therapy session, punctuated by standards such as “Hey There,” “Sisters” and “It’s Only a Paper Moon”? Or is it a nightclub act, interrupted by psychobabble?

Another issue is that either Clooney’s story exactly fits the standard biopic narrative — it could be called “Walk the Beautiful Sweet Dreams Line” — or it has been shoehorned into that template. Either way, the story feels so familiar that the Clooney character is all too apt when she sums up everything “Tenderly” has to say about her life with one little phrase: “lousy childhood, lousy marriages, drug-induced psychosis.”

Fortunately, “Tenderly” is as much a concert as it is a play — and the music works spectacularly. Backed by a zippy, three-piece combo, Anderson whips through Clooney’s hits, as comfortable with a big, achy torch song like “I Get Along Without You Very Well” as she is with the speedy and enjoyable kitsch of “Mambo Italiano,” which earned the biggest ovation on opening night. Anderson has style, which is saying something for a woman who has to perform the whole show in a shapeless skirt that vaguely suggests a poopy diaper.

The show has style, too. From Erik Paulson’s handsome set to Eric Morris’ brisk direction, which makes the most of Anderson and Shipley’s talents, it’s an entertaining evening at the Old Log, a venue that stretches back to 1940, six years before Clooney made her first recordings.