Keep all leads short and away from 60 Hz power wires to minimize AC and electrostatic DC pickup. Place a grounded aluminum sheet on the workbench, and under the DVM and other components to further reduce spurious pickup by the wiring. A piece of grounded kitchen aluminum foil will do nicely for the aluminum sheet. You may find that the reading of Vd slowly drifts upwards. Wait it out.
Most diodes exibit a photoelectric effect that can cause measurement error.
Guard against it. Here are some Germanium diode measurements made using a Mathcad program written by Ray Waugh of Hewlett-Packard. For Schottky diodes, the value of n and Is seem to be independent of diode current. Multiple similar diodes may be paralleled to increase Is. Is is increased proportionally to the number of diodes in parallel.