Well, Sourdough, Ray et al, if the song was covered by Big Mama Thornton three years before Stoller and Lieber copyrighted it, it seems very likely it was around in the black community before they took it over. Seems like her words are a little different too. I'll go give it a listen.
Possibly it isn't as simple as "stealing" either. There were some very different attitudes toward song authorship in the 50's. People who discovered or collected the songs felt justified in putting their names on the sheet music. Witness the 3 people who "authored" Malvina's "Little Boxes" or "Turn Around".
Perhaps L&S both wrote songs in the style of black singers, and collected songs from black singers, and put their imprimatur on both.
Blessings,
Barbara