In early 2022, Jander Durán, a hair stylist in Colombia, fled the country with his wife and young daughter after a guerrilla group that had moved into their village made it clear their lives would be in danger if they stayed. Mr. Durán decided their best hope was making a case for asylum in the United States.

They made their way to Texas, and were quickly overwhelmed with the complexity of proving their case: They would need concrete proof of the threats they had received and evidence of Mr. Durán’s father’s political activism, which the family believed was the reason for the threats. The process would require hundreds of pages of paperwork. Most immigrants founder there; more than 80 percent of asylum cases are rejected.

Knowing that losing their case could be a death sentence, the Duráns hired an immigration lawyer in San Antonio to help them through the process. But on the day their case was finally called before an immigration judge in January, their lawyer made only a brief appearance on a video screen and notified the judge that he would no longer be representing the couple — they had not been able to agree on his fee.

Mr. Durán, who said he had been unable to raise what the lawyer was asking for, looked at his wife in confusion and disbelief. The judge postponed their case to give them time to come up with a solution.

Migrants fleeing violence and poverty from all over the world have crossed the U.S. southern border in record numbers in the past few years, a surge that has overwhelmed shelters, left cities scrambling for resources and added to backlogs in the immigration courts. The surge has also created a new obstacle for migrants hoping to win asylum in the United States: a serious shortage of lawyers to help navigate the notoriously complicated legal process.