In an ornate room in Georgia’s Capitol, Julie Adams — a member both of the election board serving the state’s most populous county and of a right-wing organization sowing skepticism about American elections — got the news she was waiting for. And she couldn’t wait to share it.

With pink manicured nails that matched her trim pink blazer, she tapped out a message on her phone to a top election lawyer for the Trump campaign and the Republican National Committee. “Got it passed,” she wrote to Ginee...