AUSTIN, Tex. — For nearly a decade, meetings of the Republican Governors Association were buoyant, even giddy, affairs, as the party — lifted by enormous political donations and a backlash against the Obama administration — achieved overwhelming control of state governments.

But a sense of foreboding hung over the group’s gathering in Austin this past week, as President Trump’s unpopularity and Republicans’ unexpectedly drastic losses in elections earlier this month in Virginia, New Jersey and suburbs from Philadelphia to Seattle raised the specter of a political reckoning in 2018.

“I do think Virginia was a wake-up call,” said Gov. Bill Haslam of Tennessee, who took over here as chairman of the governors association. “There’s a pretty strong message there. When Republicans lose white married women, that’s a strong message.”

In a series of closed-door meetings, governors tangled over how best to avoid being tainted by Mr. Trump, and debated the delicate task of steering Mr. Trump’s political activities away from states where he might be unhelpful. Several complained directly to Vice President Mike Pence, prodding him to ensure that the White House intervenes only in races in which its involvement is welcome.