According to the amazing Worldwide Was fan site:
"Dignity - Sweet Pea Atkinson & The Was (Not Was) Quintet - September 21st, 2004. Sweet Pea on vocals, David on harp, Don on string bass, Wayne Kramer on guitar, David McMurray on sax and, possibly Randy Jacobs on guitar and Narada Michael Walden on drums. From a forthcoming album that consists of covers of Bob Dylan songs performed in the style of early 50’s Chicago blues." I don't hear a sax but I do hear piano. Wonder who's tinklin'? (And if the album was "forthcoming," it still is!)
Bob Dylan originally recorded the song for 1989's Oh Mercy. A few (though not all) versions are online here.
"They moved on to the next song, "Dignity", which was recorded with Dylan, Stoltz, and Green. Though they managed to complete a polished performance, Lanois suggested something more ambitious with a Cajun band. Curious to see what Lanois had in mind, Dylan agreed to recut the song. The next evening, a session was held with Rockin' Dopsie and His Cajun Band, but the results were disastrous. The group experimented with different keys and tempos, but according to Dylan, everyone was frustrated with the results. Dylan still preferred the original version recorded the previous day, but it wasn't considered finished by Dylan or Lanois. (In his autobiography, Dylan refers to the original version as a "demo".) As the session continued into the early morning hours, the group gave up and began playing old standards like "Jambalaya", "Cheatin' Heart", and "There Stands the Glass". It was during this time that Dylan tried out another new song, "Where Teardrops Fall". "I showed it quickly to Dopsie and we recorded it", Dylan later wrote. "Took about five minutes and it wasn't rehearsed."
The next day, they listened to every take of "Dignity" recorded with Dopsie and his band, and all of them were rejected. "Whatever promise Dan had seen in the song was beaten into a bloody mess", Dylan recalled. "Where we had started from, we'd never gotten back to, a fishing expedition gone nowhere. In no take did we ever turn back the clock. We just kept winding it. Every take another ball of confusion.
...Another outtake, "Dignity", was one of the first songs written for Oh Mercy. Dylan viewed "Dignity" as a strong contender for the album, and an extensive amount of work was done on it. However, Dylan was dissatisfied with the recorded results, resulting in his decision to omit it.
...Easily the two most celebrated outtakes from Oh Mercy's sessions, Dylan would not only perform "Dignity" and "Series of Dreams" live, he'd eventually issue them on official releases... "Dignity" was performed live during a 1994 appearance on MTV Unplugged, and the same performance was later issued on the accompanying album. A radically remixed version of "Dignity" featuring new overdubs was released on Bob Dylan's Greatest Hits Volume 3, while the original Lanois production would not see release until the soundtrack album of the television show, Touched by an Angel."
(Interestingly, Dylan's next album, Under The Red Sky, was produced by... Don and David Was.)
Dylan's demo version and an alternate take are on the 2008 collection Tell Tale Signs: Rare and Unreleased 1989-2006 (Bootleg Series Vol. 8).