On shore power, the switch should be "closed", which means the converter can charge the battery. If the switch is "open", the battery cannot run the 12v things when the converter loses its 120v input.
The battery being "dead" needs clarification. Low voltage at the battery posts using your digital multimeter? Or just no 12v in the rig now there is no power to the converter and the disconnect is open?
There is also a battery fuse or DC breaker near the battery on the pos side that could have blown (to open) that would have the same effect inside the rig as if the disconnect were open.