AMMAN, Jordan — For the dozens of Syrian and Iraqi refugees who gathered at the international airport in Amman, Jordan, to board flights to the United States early Monday morning, the wait was finally over.

Some had prepared as long as four years for this moment. Others had lingered for months in camps, jobless and unsure about the future. Some had arrived with children, including children with disabilities. They had experienced waves of despair and, even now, were skeptical about whether their deliverance was at hand.

When a 27-year-old Syrian man who gave his name as Osama found himself waiting to board a flight to Chicago on Monday morning with his wife and four children — including a young disabled daughter — he was overcome with emotion, but he was also angry.

He said his family had been waiting 18 months to board a flight for the United States, and then President Trump’s travel ban quashed their dreams. Like many of the refugees who had gathered at the airport on Monday, he said he could barely believe that the ban had been challenged, and that he would soon be stepping on American soil.

“Let’s see, hopefully. Let’s see, God willing,” he said, looking pained and at least a decade older than his real age. “I hope so. Let’s see.” He was nervously smoking a cigarette outside the terminal, as his wife — whom he had told not to speak to journalists — waited inside.

Osama, who asked to be identified only by his first name because of concern for his family’s safety, said his life in Jordan had not been easy since he had fled the Syrian civil war and his home in Homs in western Syria in 2012. After arriving in Jordan, he had shuttled from camp to camp, ending up in Ma’an, an impoverished southern town.

He barely had enough money to survive, he said, and hoped that life in the United States would be easier, especially for his daughter, for whom he had struggled to get adequate care in Jordan.

Others who were waiting were filled with hope that they would find the “American dream” at the end of their journey, even under a new president who has made clear his disdain for refugees. The travel ban — now the subject of battles in federal courts — suspended the entry of Syrian refugees indefinitely and of other refugees for 120 days, and it temporarily barred entry into the United States by visitors from Iran, Iraq, Libya, Somalia, Sudan, Syria and Yemen.

One couple, Hasan Alwan and Sanieh Azzawi from Baghdad, who were taking the flight to Chicago, said they had waited two years to travel to their final destination: North Dakota.

“I am excited,” Ms. Azzawi said. “We have been waiting for this day, and we were so afraid it would not happen,” she said, referring to the ban. “People who I know in the U.S. told me, ‘In America you follow the rules and everything will be all right,’ but we became confused about the rules. Now we are going to America and life will be good for us.”

Fawzah Abdul Haseeb and Ibrahim al-Ahmad from Homs were heading to Austin, Tex., via Chicago, with two disabled children, and they said they had been scanning the newspapers every day to see if the travel ban had been lifted. They said they did not know anyone in Texas and wondered if Austin was a rural or urban area. Either way, they were hoping for a life different from the one they had found in Jordan over the past four years.

“My daughters need help, and they can’t receive it here,” Mr. Abdul Haseeb said. “We need to work. Goodbyes are difficult. We will miss our family here. We are far from our homeland. But we are going where there will be a future for us. We are going for something better for us and for our children.”

Reem Arash and her husband, Khalid, from Damascus, Syria, said they were bracing for culture shock in their new home in Ohio. But Mrs. Arash, who had lived with her three children in a Jordanian refugee camp while her husband tried to earn money in Saudi Arabia, said she craved what she saw as the safety and security of the United States.

“It was very difficult for me; the camp was chaotic and unsafe, especially that I was on my own without my husband,” she said as she waited for the Chicago flight. “We lived in a tent, but I was scared to go at night to the bathroom or to leave my tent.”

After her husband had joined her in Jordan, she said, the two moved to the town of Zarqa, a gritty, working-class city northeast of Amman, where they were unable to make a living. After registering with the United Nations refugee agency, she said the family received food coupons amounting to $70 a month, barely enough to feed her three boys, ages 8, 9 and 12.

She said news of Mr. Trump’s refugee ban had been devastating. When the telephone rang on Sunday, and an aide worker told her to be at the airport by 6 a.m., neither she nor the children could sleep because of their excitement, she said.

“We don’t know anyone in Chicago or Ohio,” she said, adding, “Beginnings are always difficult.”