1987 Nissan Pathfinder Idiot lights come on with no key in

I have an 87 pathfinder that has been through hell, but it still runs. Yesterday it developed an electrical problem where it wasn't charging the battery from the charging meter's point of view. I originally thought it might be a loose alternator belt, but after replacing it, I discovered another issue. The oil, battery, and emergency brake idiot lights come on as soon as I connect the battery. They are on when there is no key in the ignition switch.

I originally suspected a short in the ignition switch and disconnected the ignition switch wires by disconnecting the 5 pin plug from the ignition switch. Even with the switch disconnected the idiot lights come on when the battery is connected. I think that eliminates the switch as part of the problem.

When the engine is started (ignition switch reconnected) the idiot lights go off (except for the emergency brake which is on). With the engine running I put a VOM across the battery, and it reads 13.65 volts. It drops a little when the head lights are turned on (about 13.35). It drops to about 12.5 volts when the heater fan is turned on full blast. I would have thought the headlights would draw a lot more than the heater fan. In all cases the charging meter is slightly below the middle - which is what first made me think it wasn't charging.

Can someone suggest where the problem might be? I don't have an idea of where to look right now.