AUSTIN, Texas—Despite raising millions of dollars and attracting international media attention, Democrat Wendy Davis is struggling to make her campaign for Texas governor competitive.

Democrats had high hopes that the state senator, who became an overnight sensation among liberals last year with a marathon filibuster of an abortion bill, could turn the tide in Texas, where the party hasn’t won a statewide race in two decades. But Ms. Davis has failed to garner traction against Republican Greg Abbott, the state attorney general,...