When Melissa Marti saw the email that caused her to sit in her driveway sobbing into her steering wheel, she knew she was going marry Seth Visser. She just didn’t know when.

“We had been talking about it for years — we had even started making plans,” said Ms. Marti, who is from New Ulm, Minn.

But meeting a deadline to present a marriage certificate to the Roman Catholic school where she worked as a dance coach was not something she was willing to do. And so, in June 2018, a few weeks after she got the email that made her sob, she quit.

Ms. Marti, 40, had been coaching the Sonics Dance Team at Cathedral High School for nearly 20 years when New Ulm Area Catholic Schools officials wrote to say her contract renewal was on hold because her living arrangement was under investigation. Cohabitating, the email said, ran counter to Roman Catholic teachings about marriage being a moral prerequisite to living together. A subsequent letter offered two options for preserving her job as head coach, a position she had held since 2002: She could change her housing situation or present a marriage certificate before June 15, 2018.