"... after a move my microwave and vacuum both would not work. ..."
Microwave plug behind the micro vibrated loose?
Is the vacuum permanently installed? It may have a plug as well. Look near or behind the vacuum.
Tripped GFI in the outlet, in a nearby daisy chained outlet, in circuit breaker on main panel?
Wire connections vibrated loose somewhere in the system? Dutchman has been known to use wire nuts for 120 volt circuits. Wire nuts are OK for stationary use. Not so good for mobile use.
"Also with furnace running and lights on the lights will Dim and the blower fan slows down. When this happens I can hear another fan kick on I'm assuming on the invertor. When that fan goes off the lights get bright and the furnace fan speeds up. If I turn the lights off it seems to quit. I have 123v at both outlets and 12.6 volts at the battery when plugged into shore power. ..."
Furnace fan and lights run on 12 volts. The second fan starting up means converter is converting more 120 volt line power to 12 volt power.
There may be other devices using 12 volt power. Refer, battery charging, etc. Since batteries are at 12.6 volts when using shore power, they may be drawing the full remaining capacity of the converter to charge. When the furnace fan starts, the voltage drops because the converter is maxed out.
Wait 6 hours and recheck the battery voltage. The voltage should rise significantly if they are charging properly. After 10 hours from the time you plugged into shore power, the current draw from charging should be much lower, maybe 5 amps. The dimming effect will probably not be evident then.
It takes 14 to 18 hours to fully charge a lead acid battery. Current drops as the state of charge increases. Deeply discharged batteries can draw high current. This will max out the converter and it will limit current. Voltage will fluctuate with load.
After a few hours of charging, current draw drops and terminal voltage raises. When voltage reaches about 13.6 (some chargers go to 14.4 volts), the converter switches from maximum current to voltage regulation. Thus dimming effect will disappear.
I wish you good luck and happy trails ahead!