I just looked over my half dozen different Country playlists, and I really don’t have sad country songs, besides Caroline, on any them.
Probably because I don’t care for songs about neglected or abused children or the death of a child or a dog, which is why I don’t listen to songs like Concrete Angel or Old Shep.
I do have a short song called Sam by Sturgil Simpson on a playlist, which is about a man missing his favorite dog, but it’s not a very sad song.
My step-father usually had his car radio tuned to a country station and he was fond of this sad trucker song, which is more of a sad story than a song: