I think Ed's message solves the origins mystery definitively. It's verified by Smithsonian Folkways. Here are the notes on the song from their Best of Broadside collection:

Ernie Marrs and the Marrs Family (b. 1932-1988)

Ernie Marrs was a prolific songwriter and a frequent contributor to Broadside. It was claimed that he wrote 15,000 songs (Sing Out! 43/1:28, 1993). Marrs was born in Oklahoma but later settled in Atlanta, Georgia. During his career, he worked as a stonemason, served in the navy, and was a migrant worker (Broadside issue 2).


---


The Best of Broadside: 73. "Plastic Jesus"

This was one of the most heavily debated sings printed in Broadside, as it had been earlier when it was published in Sing Out! (14/2, 1964). Its publication in Broadside led to a barrage of angry letters from subscribers asking, "How can you publish something so blasphemous?" People canceled subscriptions and claimed they would never read the magazine again. In Sing Out!, articles were written defending the song. Gordon Friesen replied, "'Isn't the song sacrilegious?' We ask in turn, 'Where does the sacrilege lie really, with the song, or those greedy for profits, who debase the Savior by producing and peddling these cheap little trinkets in his image?'" (Broadside issue 39). Indeed, the song still provokes outrage from those who fail to recognize that its target is the purveyors of religious kitsch, not religion itself.

Marrs arranged his version based on a song that was already in circulation, although it was frequently credited to him. Actually, it was written by two West Coast musicians, Ed Rush and George Cromarty, who were members of the Goldcast Singers. Ed Rush traced the song back to an African-American camp-meeting song with lyrics "I don't care if it rains or freezes, leaning on the arms of my Jesus," which was the theme song of a religious radio program broadcast from Baton Rouge in the 1940s (Broadside # 41). The parody lyrics are based on the line quoted above. Folklorist Richard Reuss found six variants of the religious words in the Folklore Archives at Indiana University (Broadside #41).

Text and quotes extracted from the notes by Jeff Place accompanying The Best of Broadside.

Smithsonian Folkways recordings featuring Ernie Marrs: Broadside Ballads, Volume 6 - Folkways 05315; Broadside Ballads, Volume 5 - Folkways 05312; We Shall Overcome: Documentary of the March on Washington - Folkways 05592

---

I'd sure like to see the lyrics for the religious song. Anybody have Broadside #41?
-Joe Offer-