The city’s police chief, Art Acevedo, said the city had handled 56,000 calls to 911 since 6 p.m. Friday and had rescued 2,000 people. He said the city was aware of about 185 requests for rescue that had not yet been handled.

“A lot of people are frustrated,” Chief Acevedo said. “Hopefully today we’ll get to the rest of you. Please don’t give up on us. None of us are going to give up.”

In Houston, harrowing close calls shook many families.

Maya Wadler, 17, recalled the moments before she was rescued from her home Sunday around 4 a.m. The water, she said, “bubbled up from the doors, seeped in from the windows.”

“Everywhere you turned,” she said, “there would just be a new flowing puddle. It just kept filling. It passed the outlets. I was so scared. We didn’t know what would happen. And there is no one you can call.”

Ms. Wadler was eventually helped onto a dump truck driven by rescue workers.

“I was sitting in the corner holding my dad really tight,” she said. “I usually just trust my parents that everything is going to be O.K. But I looked up and I saw that my dad was closing his eyes, the water was getting in his eyes. And I just thought: He has absolutely no idea where we are going to go.”

For a vast swath of southeast Texas, there may be more trouble in the days ahead. The National Weather Service said the storm was expected to linger for a number of days. It predicted an additional 15 to 25 inches of rainfall along the upper Texas coast and southwest Louisiana through Friday. The service also raised the possibility of 50 inches of total rainfall in some areas, exceeding previous Texas records.