Little Talks - Of Monsters and Men is, lyrically, a conversation between two people.
One Bourbon, One Scotch, One Beer - George Thorogood is like a dude telling a story, with a bit of song in between the story.
Same with Alice's Restaurant Massacree by Arlo Guthrie, but a funnier and longer story.
I know a lot of love songs that sound like they are someone talking to a person, which is what I thought of seeing Jolene in your list. I don't think what you're after is a specific genre though, and given the examples I could probably just list every single song I have favorited on Spotify that isn't purely instrumental.