By mid-1984, Texas honky-tonk heartthrob George Strait was on a hot streak at country radio. Since his first Number One hit, “Fool Hearted Memory,” in 1982, he’d had four more chart-toppers, all of them in a row. Thirty-five years ago today, on June 25th, 1984, he was at Nashville’s Sound Stage recording studio on Music Row, recording another. But had circumstances been a bit different, the song could have been a major hit for the artist Strait first heard singing it.

Although the cities of Dallas and Fort Worth are just 30 miles apart,...