Riverman -- the only R&B song I know like that is "Beware" by Louis Jordan, which of course would have been earlier than 1960, but somebody could have remade it. I'm probably not being much help here.
A 1963 R&B (sort of) favorite of mine was "I Know, I Know" by Pookie Hudson. Also Baby Washington "That's How Heartaches Are Made."
My favorite Baby Washington opus from 1963 is "Leave Me Alone," in which she is backed by a full orchestra. I don't believe this record even charted.
I downloaded the Beware by Louis Jordan and thats not the song.I only heard this song once-at a junior high dance in 1960.It was one of the most haunting songs I ever heard.It was an up tempo tune with incredible vocals.
1963 was a fabulous year for music.Fingertips by Stevie,Heat Wave by Martha and the Vandellas,Hello Stranger by Barbara Lewis,Precious Words by the Wallace Brothers and Groovy Baby by Billy Abbott.