“I suspect that September will be a test for all of us,” said Representative Charlie Dent, Republican of Pennsylvania and a senior member of the Appropriations Committee. “We have to demonstrate to people that we have a capacity for governance and for helping our fellow citizens who are just in dire need.”

Above all else, time is running short.

The Treasury secretary, Steven Mnuchin, has urged Congress to raise the debt limit by Sept. 29. Republican leaders are intent on doing so — the Senate majority leader, Mitch McConnell of Kentucky, said last week that there was “zero chance” that Congress would allow a default on federal debt. But they risk rankling conservatives who have pushed for any debt-limit increase to be paired with spending cuts.

Days after that deadline, Congress needs to pass a spending measure by Oct. 1 to keep the government open. Speaker Paul D. Ryan of Wisconsin acknowledged last week what seems inevitable on Capitol Hill: Lawmakers, he said, will likely need to pass a stopgap spending measure in order to avoid a shutdown.

Such a measure would be a disappointment for Republicans who promised a return to the orderly funding of the government. But a Republican in the White House has not eased that process. Last week, President Trump threatened a shutdown unless Congress funds the wall that he has promised to build along the southern border with Mexico.

Republicans also must reauthorize the National Flood Insurance Program — which is currently $25 billion in debt to the Treasury and facing a wave of new claims from Harvey — and provide funds for the Children’s Health Insurance Program.