With the starter engaged, voltage will be down a bit, because of the load on the battery from the starter, although 10V seems a little low. It's also possible that there's some crud or pitting on the connection to the small terminal inside the switch that causes a voltage drop. I would start by cleaning up the cable connections at the battery, frame ground, engine ground, and starter, and see if things improve. You can also track down a bad connection by comparing voltages (under load) between say battery post itself and the cable clamp, and do that down the line toward the starter.
