A faction of local, county and state Republican officials is pushing lies, misinformation and conspiracy theories that echo those that helped inspire the violent U.S. Capitol siege, online messaging that is spreading quickly through GOP ranks fueled by algorithms that boost extreme content. The Associated Press reviewed public and private social media accounts of nearly 1,000 federal, state, and local elected and appointed Republican officials nationwide, many of whom have voiced support for the Jan. 6 insurrection or demanded that the 2020 presidential election be overturned, sometimes in deleted posts or now-removed online forums. “Sham-peachment,” they say, and warn that...