I would check your frame ground or maybe run a temporary ground from the battery to the charge/controller.
As far as your question on why the voltage drops under load. I would guess one of two answers.
1) Resistance. If there is high resistance you can get normal voltages unloaded. When using a digital meter it takes next to nothing to read 12 volts on an unloaded circuit.Think a single strand of wire the diameter of a human hair will read 12 volts with a digital meter.
2) Your load center might be failing and not passing the current necessary while under load.
Maybe try tapping into the socket holding a bulb that dims. Then when the bulb gets dim check and see if you have the same voltage at the bulb as you have at the charge controller.
Last thing. Pay attention to battery voltage or charger voltage. Remember 12.6 volts and 12.3 volts are vastly different. So when plugged in you should get the same voltage or voltage within a tenth of a volt at the battery, at the controller and at the light bulb.
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2001 Suburban 4x4. 6.0L, 4.10 3/4 ton **** 2005 Jayco Jay Flight 27BH **** 1986 Coleman Columbia Popup