I'll reckon you haven't heard one of these in a righteous spell:
Betty Everett, You're No Good, 1963. More soulful than the 1975 Linda Ronstadt version.
The Ikettes, I'm Blue, 1962.
Lee Dorsey, Ya Ya, 1961. Simplicity itself. Three chords. I-IV-V progression. No bridge.
Paul Anka, A Steel Guitar and a Glass of Wine, 1962.
Carole King, Crying in the Rain, 1963. The earnest girl-feeling of young Carole makes it better than the Everly Bros.' more polished and better executed version.
Don Gibson, Sea of a Heartbreak. A crossover hit from 1961. It's a crime for the oldies stations to ignore this great song.
Ketty Lester, Love Letters, 1961. Gets some play, but not enough.