DALLAS—Democrats from across the country passed through Texas over the weekend with dreams of reaching the White House, a stark reminder for Republicans that an energized Lone Star State Democratic electorate could help the party take back the presidency next year—and potentially for a generation.

“I’m looking for some blue Texans,” Sen. Elizabeth Warren (D., Mass.) told a crowd of about 700 people here on Sunday. She told reporters afterward that Democrats can flip the state into their column in 2020 because “people in Texas...