The source is credited in the collection: "Locke, Eleanor G., American Folk Songs for Teaching. Unpublished collection, Holy Names University. Collected by Betsy Moll." Who was Betsy Moll? Betsy Moll was one of the originators of the Kodály method in the United States. She studied in Hungary and had many professional successes, including starting pre-school music in the Pittsburgh, Pennsylvania area. She received an OAKE lifetime achievement award in 2014 and has also received an award from the Hungarian government.
Mrs. Hale's Harmonies theorizes that Moll collected this song from Hungary, which is a theory I saw echoed in other places on the internet (Google AI repeated it to me when I was searching the song history.) As I was Googling Betsy Moll, I gathered my courage and messaged her on Facebook Messenger.
Thus, "We Are Dancing in the Forest" remains a mystery of music education - it probably was created by a teacher in New Jersey, but we may never know!
I play this game by choosing one student to be a wolf and having the other students "dancing" through the room while singing the song. At the end of the song, they ask "Mr. Wolf, Mr. Wolf, are you there?"