This problem has indeed to do with the model of your plane, and more precisely with lamina faces, faces that have been welded improperly together. Resulting in 2 faces forming a box with no volume. Look at the inside of your plane if you see polygons which face the wrong direction, if you can find those, delete them and check if this does not delete parts of the actual plane, and test if it reduces the amount of shadow leaks.

The general way to fix this problem with aircraft is deleting all bottom polygons of the planar parts of the aircraft and then enabling the 2-sided option in the W3D options for those planar parts. This will remove most, if not all of the shadow leaks.