(CNN) — As freezing temperatures from a coast-to-coast winter storm arrive in El Paso, Texas, some migrants who haven't turned themselves in to border agents or officials after crossing the US-Mexico border are having a difficult time finding shelter.

El Paso is in the midst of a declared state of emergency over thousands of migrants living in unsafe conditions, as a Trump-era border policy keeping migrants out of the US remains in flux amid court proceedings.

The city has opened government-run shelters at its convention center, hotels, and several unused schools, but is unable to accept migrants who don't have documentation from Customs and Border Protection, said El Paso city spokesperson Laura Cruz Acosta.

The city must follow state and federal policies, which she said require migrants to have documentation in order to receive shelter at government-run facilities.

If undocumented migrants show up at government-run sites, they'll be connected with Customs and Border Protection to start the process of turning themselves in or are connected with shelters run by NGOs on the ground, she said.

Three men, who did not want to be identified, told CNN they have been expelled from the US multiple times in recent weeks and no longer want to turn themselves into border authorities because they have been refused legal entry so many times. The men say they crossed the Rio Grande and passed through border barriers without being detected by border agents.

As dangerously cold temperatures arrive this week, US border officials are warning migrants seeking to enter the country of the dire weather.

"Extremely cold, below freezing temperatures are expected along the Mexico and United States border during the next several days," Hugo Carmona, Acting Associate Chief of US Border Patrol Operations, said in a video statement. "Do not risk your life and that of your loved ones trying to cross the river or the desert. Help avoid human death and tragedy, stay home or remain in a safe shelter. This is a warning of extreme importance."

Shelters are over capacity

Migrants without documentation have few options for shelter, but some NGOs are accepting them into their facilities. CNN spoke with two shelters who are accepting undocumented migrants but are experiencing overcrowding.

John Martin, deputy director of the Opportunity Center for the Homeless, told CNN he's asked the city's emergency department to open the convention center to undocumented migrants for at least the next two days but has been unsuccessful, he said.

More than 290 men slept at a men's shelter run by his organization Wednesday evening, despite the space being designed for just 125 people.

"We are literally having to close our doors tonight," he said. "We just can't fit anyone in."

At nearby Sacred Heart Church, officials said their normal capacity is 130 but are expanding to 200 for the next four nights and are prioritizing women and children, they said. But hundreds of people continue to wait outside the church.

"My biggest fear is somebody is not going to wake up tomorrow morning," Martin told CNN. "I hope that I am absolutely wrong."

Many of the migrants who are coming into El Paso are not looking to stay, said Mario D'Agostino, El Paso's deputy city manager, but the city's infrastructure was struggling to support the crowds pouring in and trickling out.

"We do have a moderate-sized airport, we have a couple of smaller bus terminals, but that's not enough to keep up with normal holiday traffic," D'Agostino told CNN's Boris Sanchez on Saturday.

Now on top of the traffic, hundreds of migrants are looking to leave the city daily. "We don't have the infrastructure — the flights out of El Paso, the buses out of El Paso — to keep up with this flow."

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