After Thabit al-Qadhi was kidnapped in 2006 by Qaeda terrorists and eventually released, he said, he reported details about the episode to American officials in the hope that they would be caught. And on his way home from the Baghdad base one day, he came upon four American troops injured along the road, and he said he had loaded them into his car and driven them to the Green Zone. “America has abandoned its responsibility to protect those who protected and cooperated with the Americans,” he said. “It’s a decision solely based on my religious faith. It’s discrimination solely on religious grounds.” He added: “This is the wrong decision. Is it even constitutional?” The family’s home was along one of Baghdad’s major highways, an entry point for American troops during the invasion. The family huddled in a windowless section of the home for 10 days, as bullets shattered windows and rockets blazed through the sky, until the United States took control of Baghdad. In the years after the invasion, the family remained in Iraq, even as unrest spread, militants took up arms and centuries-old tensions between Sunnis and Shiites flared up. In addition to Thabit al-Qadhi, Ms. Obaidi’s husband was also kidnapped, on two separate occasions. The family members recalled the lengths to which they went to get them freed, how they stuffed $60,000 in Iraqi dinars into garbage bags and were instructed by cellphone to travel to a series of locations before dropping off the ransom. “It was like in the movies,” Ms. Obaidi said in an interview this month. Fearing more attacks, the family left Iraq for Jordan. It joined many other Iraqi refugees, including extended family members. Ms. Obaidi and her husband used their savings to buy a home in Amman. Employment opportunities were scarce for Iraqis, leading three of her sons, starting in 2010, to venture to the United States to find work. When Ms. Obaidi later followed them, she hoped that her entire family could apply for asylum and unite in America. She made the trip despite a number of concerns. “At first, I felt afraid,” she said. “How can I live in this country? It is a foreign country. It is very far from my culture. How will I be compatible with the community?” To her surprise, Ms. Obaidi found New York to be unlike its gruff stereotypes. People smiled as she walked down the street. Men helped her haul heavy bags up stairs. Others offered her seats on the subway. “Everybody in America is very nice,” she said. “They are very polite, helpful people, nice people, always with a smile on their face. That is my experience.” She has found additional support from the International Rescue Committee, a global humanitarian aid, relief and development nongovernmental organization based in New York. Founded in 1933, the organization is the newest organization supported by The New York Times Neediest Cases Fund, and the only one of the eight groups whose work extends beyond the New York area. It operates in 29 cities in the United States and in more than 40 countries. The organization was instrumental in helping Ms. Obaidi adjust to her new life and connecting her with a number of social services, including health insurance and food stamps. It helped her obtain a Social Security card, navigate New York’s streets and understand its transportation system. The group also helped her study for her driver’s license permit exam, among other services. “I feel I am not lonely,” she said. “I have somebody. I have somebody to support me.” She shares a home in the Astoria neighborhood in Queens with two of her sons, Saif al-Qadhi and Qaed al-Qadhi. Her third son, Tameem al-Kadhi, and his wife, Melissa Forstrom, also live in the neighborhood. “I have a nice life,” Ms. Obaidi said. “Even though my apartment is small, I feel happy in it. I like it.” But it is a home with some notable, painful absences. “I’ve divided myself,” Ms. Obaidi said. “Some part is there in Jordan, and some parts …” She trailed off, overcome with emotion. In Amman, Thabit al-Qadhi, her oldest son, lives with his wife and 5-year-old son across the street from Ms. Obaidi’s husband, Husham, and their youngest son, Omar al-Qadhi. They have been denied asylum in the United States. In October 2015, Husham al-Qadhi was sent a conditional acceptance letter for asylum in the United States. About a year later, he received a second letter, denying him resettlement. Omar al-Qadhi, who works at Unicef to help provide water, sanitation and hygiene to Syrian refugees in Jordan, was also denied resettlement. “We have lived here as if we were waiting for something, as if everything was temporary, but now we no longer know what we are waiting for,” Omar al-Qadhi said in his apartment in Amman. Thabit al-Qadhi, who has traveled to New York several times to visit his family, has not received the same denial letters for resettlement in the United States. His tourist visa was renewed, but a week later, an officer at the American Embassy in Amman told him that his visa had been canceled, and his case for resettlement was denied. Whether they will ever get approval to move to the United States is even more uncertain now. Thabit al-Qadhi said that Mr. Trump’s order was particularly painful and that he felt America was turning its back on Iraqis who had risked their lives to help soldiers during the war. “At the end we realized we were no longer welcome, neither from the Iraqis because we worked with the Americans, nor from the Americans because we were Iraqi,” he said. Like many Iraqis, Thabit al-Qadhi is living in Jordan on a conditional basis. He must renew his permission every year, and it is dependent on the family’s financial means. “Nothing is guaranteed in business,” he said. “Today, my trade company here is successful, but if one day the business fails, then what will happen? Where do I go?” In the United States, his siblings, even with employment and legal immigration status, live in a similar state of unsteadiness and concern. For three years, Tameem al-Kadhi’s only proof of his legal status was an arrival-departure record known as an I-94 form, a document without a photo of him and only his name and identification numbers. It has hindered his attempts to travel even within the United States. Visas for Saif al-Qadhi’s wife and children were approved recently after a wait of more than two years. They were booked on a plane expected to arrive Feb. 7, but Friday’s executive order by Mr. Trump has dashed those plans. “The kids, they grow up far from their father,” Saif al-Qadhi said. “All of a sudden, I told them ‘I’m sorry, something changed. I may not be able to see you soon.’” He and his wife are distraught by the development, which leaves them in a precarious position. In anticipation of the move, their children were taken out of their private school and the lease on their apartment in Amman is to expire on Feb. 1. “I’m watching the news every second,” Mr. Qadhi said. Ms. Obaidi’s children have been able to acclimate to their new surroundings and establish a rhythm in their lives. Tameem al-Kadhi owns a cellphone store in the East Village in Manhattan, Qaed al-Qadhi works as an information technology manager, and Saif al-Qadhi is an Uber driver. Ms. Obaidi stays busy by volunteering at the Masjid Dar Al-Dawah mosque. The family gathers as often as work schedules and other responsibilities allow, most often on Sundays, when Ms. Obaidi prepares a large meal. They all await the day when more chairs can be placed around the table. Halfway across the world, the other half of the family shares that sentiment. Omar al-Qadhi said he missed his brothers, but especially his mother. “It just feels weird that we are now split, and the future looks grim,” he said. “We are travelers on a journey with no destination, and my family is so far away.” Correction: January 28, 2017 Because of an editing error, an earlier version of a picture caption with this article misstated the living arrangements of two of Maha al-Obaidi’s sons. Tameem al-Kadhi lives nearby, not with her. Another son, Saif al-Qadhi, lives with her and her son Qaed al-Qadhi, not nearby.