A dunnage conversion machine converts a sheet stock material into a dunnage product that is relatively thicker and less dense than the stock material. The conversion machine includes a conversion assembly that draws the sheet stock material therethrough and randomly crumples at least a portion of the sheet stock material. Before severing a discrete dunnage product of a desired length from the substantially continuous length of sheet stock material, the random crumpling is minimized or eliminated in an area to be cut to reduce or eliminate the production of scrap shards of sheet stock material during the cutting operation.