HOUSTON — A week after Texas was slammed by Hurricane Harvey, this region was still engulfed in crisis on Saturday, with weary residents of Houston searching for ways to repair swamped homes and salvage possessions even as others faced new orders to evacuate. In cities and small towns to the east, thousands of others remained stranded by rising waters and were still without dry shelter.

After seven days, frustration and exhaustion had set in for many. Parts of Beaumont, a city of nearly 120,000, and a vast array of towns east of Houston were cut off from one another, and coping with flooded roads, submerged homes, limited power and no relief in sight. For a third day, residents of Beaumont were going without drinking water after flooding knocked out pumps for the city’s fresh water system.

“This has been a trying week,” Amelia Nickerson said, as she and her husband hauled yet another bag of trash out of their Houston home where the waters had risen after the storm made landfall late on Aug. 25. What had been their bedroom walls were being carried out, one soggy wheelbarrow load at a time. “This was so much worse than what we expected,” she said.

President Trump visited Texas and Louisiana on Saturday, his second trip to the affected region this week. Here in Houston, he toured a temporary shelter, helped volunteers load boxes of supplies and said he was “very happy” with the a recovery that, in many places, has barely begun.

As officials were only beginning to assess the widespread damage across the region and as rescue flights and boat missions continued through parts of the state, Mr. Trump was expected to ask Congress to approve $7.8 billion for disaster relief in the coming days, and $6.7 billion more by the end of the month, White House officials said.

Texas officials said 440,000 residents have applied for aid from the Federal Emergency Management Agency, and tens of thousands of people remain in shelters. Local authorities said there were at least 50 deaths in Texas that were related or suspected to be related to the storm.