Since you're only a year older than me from the same builder I can give you some direct observations that I've had. I suspect you have a bad chassis engine battery. Pull the battery and have it load tested.
With the coach plugged, even to a 15 amp outlet, the 120VAC to 12VDC converter should have kept both batteries charged even with a heater running. If the AC power demand was too much the breaker for the outlet would have tripped. When plugged in your Battery Control Center (aka BCC) will cross connect the house and chassis batteries once the house power is above 13VDC. Will do the reverse from the alternator once the chassis battery is above 13VDC.
Check the voltage at the house batteries with the coach plugged in. If the voltage is less than 13VDC you could have a failing converter. What happened to me was the converter was weak and the voltage level would drop with demand. Noticed the lights would dim when the furnace blower came on. A failing alternator could present the same symptoms.
Also you coach is build on a Ford F53 chassis.
