So far only one traditional song has been mentioned, yet most traditional songs in the English Language are in narrative form - ie they tell a story.

For a start there are 305 Child ballads - plenty of story songs there.

Taking a few others at random iow some personal favourites

Brigg Fair and Searching for Lambs are both love songs but both are expressed in narrative form
Lovely Joan and the Highwayman outwitted about young women getting the better of predatory males
Green bushes about a young woman enticed away from her "true love" by the promise of more wealth
Van Dieman's land about transportation to Tasmania for poaching

Come on you trad song lovers, let's have a few more!