The words of “Amazing Grace”, “O Little Town of Bethlehem”, “House of the Rising Sun”, and “Gilligan’s Island” can all be sung to each other’s tunes. And the words of “The Rime of the Ancient Mariner” can be sung to all four tunes.
And the words of each of those songs can be sung to the tune of “Stairway to Heaven”. But not vice-versa.
What we’re discussing here are songs that are in alternating lines of iambic tetrameter and trimeter. It’s one of the most common meters in the language – in fact, as a category of hymn tune, it’s “Common Meter” for precisely that reason. It’s also the traditional ballad stanza.
Christina Rossetti’s Christmas carol “In the Bleak Midwinter” fits the meter, and hence the tunes mentioned, as does the old “There is a fountain filled with blood / drawn from Emmanuel’s veins” (which I’ll bet only DDG and I here know). Also the “Hymn to Joy” from Beethoven’s Ninth Symphony fits!
Grab an old hymnal if you happen to have one in storage somewhere, and experiment.
My brother-in-law sings the “Green Acres” theme song to the tune of “A Day in the Life” by the Beatles. (“I heard the news today, oh boy…”) Works great. He also does “Devil Went Down to Georgia” to the tune of “The Thing” (“get out of here with that clapclap-clap before I call the cops”). Hysterical. He’s quite a hit on the local coffeehouse circuit.