Just guessing here. Since this is an older coach with issues (your water comment) my guess is the previous owner did some "rewiring" because it makes no sense that the refer has 12v but nothing else does.
I have an 84 Jamboree MH on a Ford E-350 also. Back then most of the fridges were three ways, 12Vdc, 110ac and LP. Mine was too and the refer receives its 12 volt power direct from the house battery with an eight gauge wire to power the large draw for the 12 volt heating source for the fridge. If the power center is shut off the refer would still work as it is directy wired to the battery, pretty much everything else goes from the battery to the converter first then to the lights and furnace and so on.
I suspect as others have mentioned that the converter (the power center device that converts 110ac to 12 volts) is bad and when the power is reset it starts working again temporarily.
I replaced my complete power center and converter years ago with the new smart charger type.
With an RV of this age, it could well be that it has been modified, but as a former electrical tech, I suggest that you would be better off to get a meter and do some circuit tracing, rather than to guess. I would start at a location with known power and you will need to do this when the problem is happening.
I doubt that the amount of load has anything to do with the problem, but there is nothing wrong with trying varying loads to see if it does for some reason. As you do tests and take voltage and ohm readings, take notes and try to organize findings to determine what is common to the things which do work and to those which do not work.
Look also to the 12V fuse panel, if there is one that contains all or most of the RV fuses. Usually there will be one fuse for the furnace alone, one for all of the RV lights, probably a separate one for the refrigerator, one of the water heater and probably several others.
There is usually there is a solenoid that separates the the chassis 12V system from the coach part which is only closed when the engine is running. You should be able to verify that by use of a meter.
I would also try to draw out the electrical circuit paths as you trace things. You may be able to help in that by noting the wire colors as they usually have some kind of pattern to them. Also remember that in a 12V-dc circuit, there is usually only one wire to supply the positive side of things and the return or negative is connected to the chassis ground. A bad connection to ground can easily cause the same type of problems as would a lack of power.
The key when trouble shooting is to use some kind of system and take good notes to be sure that you don't overlook or forget something. I generally check first at the end where the item is not working, then move back to where it should receive it's power. If you have power at the supply but not at the item, next check at some point in between the two, noting if there is power or not. If there is, move toward the problem item and if not move toward the power supply.
If you do not know the route that power takes between the two, then it is best to work from the power supply toward the problem and trace the wires along the way, keeping notes and diagrams. It is unlikely that you will be able to locate a schematic for an RV of this vintage so keep all notes, even after the problem has been solved.
Grillmaster makes and excellent point that if your refrigerator is one that can operate from 12V power for cooling, it may source directly from the battery, while the lights would not. If it does so, there should still be an in-line fuse somewhere.
I had a similar problem with the 12 volt on my Oakland 5th wheel. I started chasing the circuits from the fuse panel to the place I had lost the 12 volts. The DW had a fit when she came out and found panels removed a week before we had to go to a wedding. I finally found the problem in a Scotch Lock connector used to tap into a live wire to get another feed line to another circuit ie. the outside door light. Where it had cut through the insulation to get power to the other wire the conductors had corroded and broke. Since then whenever I find any more of these connectors they are removed and a good solder and Heat Shrink connection is made. Good luck in your trouble shooting and let us know what you find
Another thing I did besides the notes was to take pictures of what wiring was behind different panels. Once I printed them I was able to use my notes to label the wires in the pictures.
That brown thing is the problem. If I press on it, it can flicker the lights. I also got the lights to fail and wiggled it and the lights worked.
It's a switch of some kind, mounted over an electromagnet. I take it that the emag is energized when on shore power or straight house battery power. The contacts in that switch are sloppy, and wiggling the piston that is drawn down by the emag causes it to work.
That is your changeover switch as you surmised. It likely has a 110v coil and the switch is the medium size version of a standard microswitch. If you tinker you can replace just the switch if you can get to it. It may be better to get a new power center/convertor. See BestConvertor.com. You may be able to replace that "relay" with one from the auto parts store and feed the coil from 12v instead.