AUSTIN, Tex. — FOR most of a long, hot summer, Wendy Davis’s campaign for governor here resembled a cowboy lost in the desert — horseless, stumbling and finally just left for dead in the remorseless Texas heat. Despite a strong national profile, she trailed her Republican opponent, Attorney General Greg Abbott, sometimes by double digits. Among the experts, a conventional wisdom set in: Ms. Davis can’t win, Republicans can’t lose and Texas won’t change. Yet as summer has turned to fall, Ms. Davis has entered new territory: Last week a poll by the Texas Lyceum, a nonpartisan, nonprofit educational institution,...