I've now discovered the phrase "foggy, foggy dew" (though not the song) in a New York State newspaper in 1870.

Troy Daily News (Aug. 12, 1870):

"A pretty awning will serve to keep the foggy, foggy dew off the ladies' frocks as they cross to and from the ball-room."

Taken together with the Alabama mention in 1868, it suggests that the song was widely known in the U.S. shortly after (and likely during) the Civil War (1861-65).