The Musical Monologue is definitely one variety of spoken word with music accompanying it. It is a genre which already has its label to identify the style of performance. Some of them might be described as a "spoken" song, but my recollection is that the musical accompaniment is more to give emphasis to particular parts of the narrative than what we would expect in a song. The question is whether the Big Bad John sort of song fits into the Musical Monologue?

Your original post is looking for a term to describe or label what you are working with. Perhaps you could use multiple descriptions (musical monologue/spoken word/spoken song/etc.)in the way a web page uses tags which a search engine uses to "find" a page.

Another Country Music spoken song is Convoy by CW McCall. The style does seem to be used more in Country music compared with other "pop" music.

Quack!
GtD.