SIKESTON, Missouri — On a Friday evening in late July, Catherine Hanaway stood before 50 attentive citizens of a small town in southeastern Missouri—men in slacks and sneakers, women with nicely done hair. Her 6-foot-tall frame swayed slightly as, with her usual frankness, she made a spirited case to be the next governor. The country is breaking down, the former federal prosecutor told the group gathered in the newly opened headquarters of the local Republican Party—police ambushed and murdered in Dallas, in Baton Rouge, Louisiana and Ballwin, Missouri—three shootings and eight dead in 10 days. As people nibbled cupcakes topped...