If you put the DVM in series with the ground circuit and read 10 volts I would think your battery is weak for whatever reason. A DVM draws very little current in voltage mode, at best a milliampere or so. On the other hand, with all circuits off, you should read zero voltage as that’s an open circuit. On your horn relay, did you remove the horn wire or the voltage side of the relay. If you removed just the horn wire, the relay could still be closed and drawing current if the ground side of the relay is grounded.
