Mr. Youngkin accepted Mr. Trump’s endorsement but hasn’t campaigned alongside the former president.

That approach has proven effective with voters such as Tywana Hampton, who lives outside Richmond. “To me, he’s like a pretty face on Trump—a person who is all of the good things and not the bad,” said Ms. Hampton, 62. She voted twice for former President Barack Obama before backing Mr. Trump, drawn by his policies more than his personality.

Republicans say that a cluster of issues involving education can win swing voters while energizing the GOP base. Mr. Youngkin has allied himself with parents upset about the handling of a sexual-assault case in Northern Virginia’s Loudoun County public schools. He has also said he would ban schools from teaching critical-race theory, though county school officials say it isn’t part of the curriculum…

Mr. Leachman, 48, said he liked Mr. Youngkin’s demeanor and business background. “I think he’s calm and cool and collected—not a polished politician but a polished human being. I’d trust him to coach my kid’s basketball team,” said Mr. Leachman, who works in sales for a moving and storage company.