For a moment in August, an event hall in Texas teemed with hope, taquitos and unity.

It was a border-town stop for Beto O’Rourke’s Senate campaign, but another Democratic politician commanded particular attention: Gina Ortiz Jones, a history-making congressional candidate — gay, Filipina-American, an Iraq war veteran — hoping to turn a majority-Hispanic district blue. “Really special person,” Mr. O’Rourke said, as Ms. Jones stood and waved.

But soon, a county chairwoman posed an uncomfortable question. Mr. O’Rourke had not endorsed Ms. Jones. In fact, he had elevated her Republican opponent, Representative Will Hurd, with frequent praise and, most memorably, a live-streamed bipartisan road trip that helped jump-start their midterm campaigns. Would Mr. O’Rourke support the Democrat?

He would not.

“This is a place where my politics and my job and my commitment to this country come into conflict,” Mr. O’Rourke said. “I’m going to put country over party.”

Some supporters of Ms. Jones saw it differently: Beto O’Rourke was once again putting Beto O’Rourke first.