I also found sources that talked about dispersal by animals, either by birds collecting it for nesting material, or by the fibers getting snagged on passing animals. I even found references that claimed that the thick bundle of fibers is an adaptation for dispersal by water. Cotton grows naturally along the coasts in South and Central America, as well as on Caribbean islands. The seeds are water resistant, and the clumps of cotton float, so it is possible.
As with many questions in science, there is probably more than one correct answer.