Each morning at 3:30, when Joann Bourg leaves the mildewed and rusted house that her parents built on her grandfather’s property, she worries that the bridge connecting this spit of waterlogged land to Louisiana’s terra firma will again be flooded and she will miss another day’s work. Ms. Bourg, a custodian at a sporting goods store on the mainland, lives with her two sisters, 82-year-old mother, son and niece on land where her ancestors, members of the Native American tribes of southeastern Louisiana, have lived for generations. That earth is now dying, drowning in salt and sinking into the sea, and she is ready to leave. With a first-of-its-kind “climate resilience” grant to resettle the island’s native residents, Washington is ready to help. “Yes, this is our grandpa’s land,” Ms. Bourg said. “But it’s going under one way or another.”