In a way it refers both to the Byrds and the First Edition.
The latter was doing its own reference to the Byrds song.
As I’ve heard the song over the years, it was clear to me
what was being talked about—Dylan, Stones, unrest, Beatles,
and all the stuff about the 50s and later the 60s. “The levee was dry” means he went to get water (representing—faith? hope? etc.) but could get nothing.
And how more obvious a line could he do than “The church
bells all were broken”?
"The Levee" was the name of a bar where he often went to tip a glass or two. When he went there and it was closed, he had to head up the road to a bar in Rye, NY. The good old boys in Rye were drinking whiskey.
drove my chevy to the levee but the levee was dry.......
just as the Chevrolet itself is a familiar icon of 1950s America. Also, given that a drive to a levee carries the suggestion of romance in a car, we can almost see him on a date here. But the date is over, the levee is dry—someone he once loved has betrayed him; something that once gave him sustenance has evaporated.