Republicans fear that Mr. Trump has relinquished his role as leader of the party, instead assuming the mantle of his own political movement. And they are bracing for an election season in which their deeply unpopular president does more to undermine than aid candidates of the party he ostensibly oversees.

“It’s a cult of personality,” said Mr. Sanford, who faces a primary challenge from a state legislator who charges that the congressman has been inadequately loyal to the president. “He’s fundamentally, at the core, about Donald Trump. He’s not about ideas. And ideas are what parties are supposedly based on.”

Such open divisions between a president and elected officials of the same party mark an extraordinary departure from modern political tradition. Even if they feuded at times with their president, lawmakers knew they could ultimately count on the White House to endorse and raise money for incumbents, because controlling as many seats as possible would serve both their interests.

But Mr. Trump’s decision to align himself with congressional Democrats this week over federal spending and hurricane relief cemented a view that he will not operate according to any such conventions. Relations between the president and congressional Republicans have frayed over the lawmakers’ failure to deliver on key legislation and Mr. Trump’s constant badgering and personal attacks against them.

The president has repeatedly assured Republicans that he will be an active campaigner for the party next year. But Republican leaders fret that he will gravitate toward candidates who share his distinctive political priorities and anti-Washington attitude, and that nominating some of the Trump-driven candidates could imperil their control of Congress. He will have spent as much time in his first year in office raising money for his own re-election as helping others.