On May 25, 2011, Kathy Hochul woke up a star. Secretary of State Hillary Rodham Clinton called her from London with congratulations. Nancy Pelosi, the House Democratic leader, effusively praised her. News outlets from France, Denmark and Japan all covered the story of how she had pulled off an upset in a Republican district in a closely watched special election. “It was an exhilarating time,” Ms. Hochul recalled. Life is more anxiety-filled these days for Ms. Hochul, who now is the most endangered Democratic House incumbent in New York State and one of the most imperiled in the country, according...