The article is from the July, 1882 issue of Harper's Monthly Magazine (65: 281-286). This should be the American ed. Alden's byline does not appear with the article - not terribly unusual for the 19th century, though I think by the '80s the practice was getting old. (It was so much more "romantic" and mysterious not to know the author's name!) It's been a long time, but I think Alden must be credited in the annual index.
*Some* Civil War soldiers (and a lot more Civil War sailors) undoubtedly knew a few verses to the shanty, but the idea that the song was a pop megahit like "Tramp Tramp Tramp" or "The Bonnie Blue Flag" is demonstrably wrong. I don't even know of any evidence that the melody alone was part of the usual repertoire.
Maybe the song entered the (western) army repertoire, at some point, through hearing sailors sing it in San Francisco Bay. Army versions, BTW, are mostly about "Sally Brown" - which is why Sandburg calls the song "The Wide Mizzourye."
It may have been mentioed above that Captain Whall, writing in 1910, claims to have heard an English school chum sing a song (not the shanty) about Shenandoah and his daughter before the Civil War. Whall always tries to be fussily accurate, so his recolection is proably correct. I don't believe this song has ever been recovered or solidly identified.
A search of American memory finds no relevant songs under "Shenandoah" or "wide" or "wild Missouri."