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I assumed they returned to Afghanistan and transfered the body to a plane.

They did. Some of the supposed pictures of Osama are from in the hanger at the Afghanistan airfield.

That's also where (as I understand it) they did all the DNA testing and so forth. The final stage in which the body was flown to the aircraft carrier was just for disposal (a decision I think was sensible -- rather than allowing "the body" to become a rallying issue for terrorists it's better just to dump it in the sea before anyone knows you've got it).

Not sure about these modified ones, but a normal Black Hawk can refuel midair.

According to Wikipedia, the standard "ferry range" of the UH-60L Black Hawk is 2,220 km (1,380 miles).