Glaad almost died in 2014. After years of fighting homophobic news coverage and working to bring about inclusion in the entertainment industry, the L.G.B.T. advocacy organization found itself losing money, scrambling to adapt to the rise of digital media and struggling to be taken seriously in Hollywood. “I was given a scary mandate,” said Sarah Kate Ellis, who was named Glaad’s chief executive in late 2013. “Fix it or shut it down.” Determined to revive the nonprofit, Ms. Ellis pursued new donors. She created a “rapid response” unit to contend with online media; Glaad now advises Twitter and Facebook on content policies. And she rebuilt Glaad’s credibility in Hollywood. Fixed. But Glaad again finds itself at a crossroads. Success has emboldened Ms. Ellis, 47, to push the organization deeper into national politics with a gutsy and potentially historic mission: to build support for a constitutional amendment that would explicitly protect lesbian, gay, bisexual and transgender people from discrimination.