Last April, a high school senior in Utah named Gabriela Merida joined a student town hall hosted by her governor. She introduced herself, noted the pronouns she used and broached the subject of mental health challenges facing young L.G.B.T.Q. people. How, she asked, did the state plan to help constituencies like hers?

One year later, the Republican-controlled Utah State Legislature has passed a measure to bar transgender girls from competing in girls’ sports, overriding Mr. Cox’s veto. His mention of his pronouns has become fodder for right-wing derision and misleading video clips. And deep-red Utah is now at the center of a new fight that is reordering the nation’s politics, roiling its education system — and, for some Americans, shaking their sense of belonging as a midterm election year unfolds.