Yevhen Chupyna, a Red Cross rescue worker, said that the scale of the disaster had yet to come into focus for many living in areas that might be flooded.

“The situation is difficult emotionally and psychologically,” he said as he helped unpack boxes of humanitarian aid. “People don’t really know what happened. They have not realized this is a catastrophe.”

With communications spotty, he said it was difficult to get accurate information about the state of the flooding. The city of Kherson straddles the Dnipro River, which has become a front line in the war, dividing the warring armies.

The western bank, which is where the majority of Kherson’s residents live and work, is controlled by Ukraine, retaken last fall after eight months of Russian occupation. It mostly sits on elevated land but there are some neighborhoods close to the river bank where flooding has already been reported. The eastern bank, controlled by the Russians, is kind of a bayou, with islands and marshes and many country homes accessible only by boat, even before the dam was breached.

Ukrainian officials, citing reports from emergency workers and volunteers, said some neighborhoods near the river were already flooded. Vasyl, 40, a factory worker who lives in Kherson, said in a brief text message that people were trying to evacuate low lying neighborhoods but that Russians were still shelling the area.

“Russians opened mortar fire as people prepared to evacuate from Ostriv,” he wrote. “They are terrorizing us.”

Alim, who reached out from Kherson via text message, said people in the lower part of the city were in a panic. “Some are moving stuff to the upper floors and roofs of their houses, while others are packing the cars and trying to leave,” he wrote.

Buses were being organized to take people from their home to the train station, but only about 30 people were registered to take the first 10-car train as of 12 p.m. local time. Mr. Chupyna said that they have hundreds of beds in Mykolaiv prepared for people who were forced from their homes.

Over 15 months of war, Ukrainian volunteer organizations have become adept at responding quickly to emergencies. But rising waters from a breached dam was a totally new challenge. Olha Napkhanenko, 40, a volunteer with the Serhiy Prytula Charity Foundation, said that her colleagues in the city of Kherson reported only about 5 percent of the city being severely affected as of noon, but that the situation could get worse.

As she prepared snacks for the children who might arrive, the Ukrainian national anthem echoed through the station hall as workers stacked supplies.

“The worst will be on the eastern bank,” she said, referring to Russian-occupied territory. “Unfortunately, we can’t help them.”

Svitlana Sitnik, 52 a volunteer from a different organization, said her aunt was in one of the towns on the east bank occupied by the Russians, Oleshky, and she was in contact with people there via a private Telegram channel. They painted an increasingly dire situation for the civilians there as Russian soldiers continued to patrol the streets and refused to provide assistance as the waters rose

The Russians announced an evacuation plan, people in the city reported, but details were scarce about how it would work.

For now, Ms. Sitnik said, it was neighbors helping neighbors in Oleshky. “Local volunteers are offering to help people get to Crimea,” she said.

But internet and cellular service was spotty and even if they could use their phones, she said, people are afraid to use them on the streets for fear of attracting the attention of Russian soldiers. “All the people are exhausted,” she said as she shared with a reporter the conversations on the secure Telegram channel on her phone. “They are on the edge. They have no rights.”