Pretty Woman, Roy Orbison
Midnight At The Oasis, Maria Muldaur (not really country, but it fits the OP)
Me And Mrs Jones, Marvin Gaye (again, not country)
Walk On By, Leroy Van Dyke (a Jim Reeves song)
One Has My Name (The Other Has My Heart) - Here’s the Jerry Lee Lewis version
This does lead us into a gray area… there are a lot of songs about clandestine affairs, such as “The Dark End of the Street” and the previously mentioned “Walk On By”… should we be including songs like that?
Wolverton Mountainby Claude King… a classic of the “her folks won’t let me see her” variety… of course, at the end, he says he’s going to go get her anyway, but I get the feeling Clifton Clowers is going to take care of him pretty easily.
Not exactly country, and not the era you are looking for, but there’s this:
It’s Sad to Belong - England Dan and John Ford Coley
Not exactly what I’m looking for but close enough that I will allow it!
Next in Line - Conway Twitty
That one’s exactly on point.
Please Help Me I’m Falling - Hank Locklin
Also on point.
They’re coming to me fast and furious now…
On the Other Hand - Randy Travis (Love you, but I’m married.)
Before the Next Teardrop Falls - Freddy Fender (Arguably. Not clear whether her ever had a thing with the person he’s singing to.)
“Just to See You Smile” by Tim McGraw, although they were together for a while
“Love Letters From Old Mexico” by Leslie Satcher feat. Alison Krauss and Emmylou Harris
“Whisky Lullaby” by Brad Paisley and Alison Krauss, which in my headcanon is about the couple in “He Stopped Loving Her Today”
Tight Fittin’ Jeans - Conway Twitty
A couple:
Johnny Cash - I Still Miss Someone
Travis Tritt and Patty Loveless - I Know You’re Married (But I Love You Still)
I can’t find a single country version of this, but what the hell. Midnight Confessions.
Leon Payne, The Blind Balladeer, wrote “They’ll Never Take Her Love From Me.” Hank recorded it, Emmylou recorded it & so did Doug Sahm, aka Wayne Douglas.
The Louvin Brothers wrote “When I Stop Dreaming.” (Emmylou recorded that one, too.)
Merle Haggard, Tonight the Bottle Let Me Down
Steve Goodman/David Allan Coe, You Never Even Call Me By My Name