GRETNA, La. — The country’s last major pockets of resistance to same-sex marriage were rapidly shrinking on Monday as officials in states across the South, citing the rule of law, softened their defiance and began offering marriage licenses to gay and lesbian couples.

Here in a suburb of New Orleans, Alesia LeBoeuf’s hands were shaking moments after she and her partner, Celeste Autin, became the first same-sex couple to receive a marriage license in Louisiana.

“I’m just speechless,” Ms. LeBoeuf said. “I never thought I’d see the day.”

Louisiana was the last holdout, the only state where no same-sex licenses were issued on Friday after the Supreme Court ruled that gay men and lesbians had a constitutional right to marry. Over the weekend, Gov. Bobby Jindal, as he condemned the Supreme Court decision, said that Louisiana would comply when instructed to by a federal appellate court. Licensing offices across the state said they were waiting for orders.