I confess I only knew "Sweet and Slow" by the Gerry Mulligan Concert Jazz Band. Checking that sleeve note now, I see it does in fact date from 1935, Fats's era.
I've never heard the Mulligan version. Didn't even know he did it.
I have it on a Verve LP, reissued by the British HMV label, simply called Gerry Mulligan: the Concert Jazz Band. Sleeve note writer, Alun Morgan, says of this track: '"Sweet and Slow" comes from the 1935-vintage musical Broadway Gondolier (music by Harry Warren, lyrics by Al Dubin), a show which contained also "Lulu's Back in Town". In the hands of the Concert Band this takes on a personality as close to Duke Ellington as the Ducal tributes which used to be presented by Charlie Barnet and his orchestra.'