If they said the area was levelled last Fall, then I would just loosen the surface and level that, rather than bring in more dirt... I would slit seed 2 directions and soak it all down good... no straw unless you're on a bad slope...
Is the frost out of the ground by you???
Now that I know that clay's texture(platelets) has nothing to do with water infiltration, percolation, or drainage,,, I wonder what does...