Why would public lands in Louisianna that are current being administered by the American federal government, continued to be administered that government after those lands are no longer in America?
Because they are not and have never been owned by the state of Louisiana. It belongs either to the United States or the people of the United States collectively (pick your preference).
And after the Spanish-American war, I believe Cuba was a territory for a small number of years. Or was it a possession?
As a condition of declaration of war against Spain, Congress passed the Teller Amendment
, which prohibited annexation of Cuba. After the war, the United States occupied Cuba until 1902 and then turned control over to the Cuban government with the exception of a naval base at Guantanamo Bay, which is rented in perpetuity. The United States did exercise the ability to veto decisions of the Cuban government, though, but there weren't US troops and we never claimed to own Cuba.