According to the New York-based advocacy organization Immigration Equality, thousands of these gay couples -- one American and one an immigrant -- leave the U.S. each year for countries where gay immigrants are welcome.
Vandiver and Velandia say neither wants to leave the United States.
"My training is to be a professor and teach political theory in an American context," said Vandiver. "I really need to stay in America to pursue my studies and my work. I don't think I should have to leave America."
Velandia said he cannot imagine going back to Venezuela, where he lived a closeted life.
"With my relationship with Josh in America, I found the love of my life. I can actually be myself," he said. "It's ridiculous to try to go back to Venezuela being a gay man. I was repressed by the culture and religious beliefs. Going back would be like going into the past."
"We expect DOMA to be defeated in a few years and be history," said Soloway. "We want people like Henry to be here."
Soloway is also helping Carry Tucker, 59, of Sacramento, Calif., who has been separated from her legal wife since they were married in Canada in 2007.
Claire Pollard, 49, is a citizen of the United Kingdom.
"We tried every legal way to get Claire into the country to live with us," said Tucker, an Air Force veteran who calls herself a patriot. "We looked into everything."
The couple was prepared to move to Canada where Pollard had applied for legal residence under the skilled-worker program, but the economy went sour in 2008.
Under Gov. Arnold Schwarzenegger, California recognizes legal same-sex marriages, but DOMA stands in their way.
They have now been separated for five years and Tucker's 18-year-old daughter is a "stranger" to Pollard.
"All of her teen years, my daughter did not have her loving stepmother in her life," said Tucker. "They are the two loves of my life, and they don't know each other. That is the most painful part for me."
Same-sex marriage is legal in Connecticut, Iowa, Massachusetts, New Hampshire, Vermont, in the District of Columbia and in one Indian tribe in Oregon.
In California, same-sex marriages were performed for five months until voters enacted Proposition 8. A federal judge has ruled the ban unconstitutional, and a court of appeals has issued a stay on the ruling, which is expected to work its way to the Supreme Court.
In New York, Rhode Island and Maryland, same-sex marriages cannot be performed, but they are legally recognized.
Groups opposed to same-sex marriage say DOMA is in the best interests of children.
"There will always be exceptions, but the definition of marriage affects all American children," said Ashley Horne, federal policy analyst for CitizenLink, a policy arm of Focus on the Family.
"A compassionate society promotes what is in their best interest, and that includes policies that would give every child a chance for both a mother and a father."
She said these couples know that legislative efforts to overturn DOMA will fail. "There's a reason they're not doing that -- they don't have the votes."
"Polls have tended to overestimate voter support for redefining marriage, but the national discussion gives thoughtful Americans the opportunity to consider the purpose of marriage -- -whether it's most valuable as validation for adult relationships, or as the only family structure that attempts to give a mother and father to every child," said Horne.