Quote: "What about loading the textures into a Skybox and then perform 90 degree turns and do GET IMAGE 4 times?""
Indeed, but the problem is the warped scenery at the edges of each image
Quote: "I tried that very technique with one of the skybox textures and it looked pinched at the edges. I thought maybe the textures were warped at the edges to make the skybox appear correctly from within (i.e. when two perpendicular sides meet at an angle it compensates the perspective). "
It is best illustrated with clearly defined vanishing points within an architectural scene:
The middle composition demonstrates the outcome. The scenery gets near, then far, then near again and so on.
Personally I would have created the clouds manually, there are many ways and tutorials demonstrating them.
Otherwise, programmatically speaking, you might be able to use a shader and image kit to implement a sine wave sampling algorithm to bring the pixels closer together where they are close, and further apart where they are far. You could achieve the same effect without programming by deforming a set of vertex dense 3D planes or using the warp to in Photoshop, or what TheComet stated about the Gimp.
Let us know how you get on, I would not mind learning how to reverse engineer a warped cubemap scene into a flat image.