As I understand your problem, the soil in front of your shed is heaving at a greater rate than the soil supporting the shed. Consequently reducing the heave of the shed (following the directions outlined above) while not addressing the door sweep area may not completely remedy the problem.
For frost action to occur three conditions must exist: 1) freezing temperatures 2)frost susceptible soils and 3) water must be present.
Eliminating any one of these conditions will greatly reduce the risk of frost heave.
The problem you are describing is fairly common at the entrance to unheated buildings where snow is routinely plowed, and is particularily common when poor surface drainage exists and the underlying soils are frost susceptable.
Granular soils are not inherently frost susceptable owing to their insignificant proportion of silt size particles. However if poor surface drainage or a high water table exists a "bathtub effect" (void spaces fill with water) can be created,causing an increase in the moisture content of adjacent frost susceptible soils and incresed heaving at the periphery of the excavation. Have you ever noticed a bump-dip-bump effect while driving over service trenches which were cut into frost susceptible soils and backfilled with granular soils?
If you elect to replace the frost susceptible soils, as suggested, be sure that the replacement area encompasses the entire door sweep area. Also the gravel need not necessarily be washed or angular, a clean well graded 0-4" sand and gravel would also be appropriate and would have the advantage of:1)less void space to potentially fill with water, 2)decreasing the depth to which frost penetrates below the fill, again owing to a decrease in voids, and 3)possibly providing filter criteria between the native soils and the fill soils, reducing your dependence on filter fabric to prevent the migration of fines.
Rigid flat lying polystyrene insulation is another possibility (major manufacturers typically provide design charts).
I dont know your logistics,i.e. size of shed interior supports etc.., but if the frost heave is fairly consistent year to year would it be possible to raise the shed a few inches?
Again, look at the surface drainage. The ground surface adjacent to the building should be sloped away from the structure at not less than 2%. If the building has gutters they should not discharge in the vicinity of the entrance door.
Good luck