I think you will do much better using a Curve object modifier than a Follow Path constraint.
To utilize this, first delete the constraint. Next, align the center point of your mesh with that of the curve (make the locations the same). Then, go to the Object modifiers tab in the Properties panel. After that, add a Curve modifier. Select the curve in the dropdown menu that you want your mesh to follow. Play with the Deformation Axis: until your mesh is aligned along the curve the way you want. Your modifier should now look something like this:
To move your mesh along the curve, move the arrow that corresponds to the Deformation Axis: you selected.
Here is a .blend file: