Ok, starting with what I feel is most critical?
Good battery supply is first step to getting it all running right as it either uses the battery power to run or it uses the 12 Volt Dc for controls on several things. One that jumps up is the generator needs the battery to start!
So this may be what happened to you. Guessing involved! Semi-educated?
When we plug in, the 110AC feeds a converter and that converter makes 12 VDC to charge the batteries and also will run low power things like lights, fans, etc.
The problem is that the older the RV, the older electronics were not built as well as later.
Really good converters are designed to put out higher current at first when the batteries may be low, as that charges them quicker. But then as the battery becomes closer to full charge, the converter needs to back the charge voltage down and finally level off at what we call float voltage which just keeps the batteries charged but not too much!
So when you find the mess and looks like the batteries boiled? Yes, that is what happens when they are charged too fast and too long! Hard on batteries and hard to keep the connections from corroding, so we often get trouble with all the things that need that steady 12VDc power!
That may explain a lot of the troubles you have seen and they likely got more and got worse as the batteries went downhill! Things like the monitor panel use the 12VDC so they can get "goofy"..
This may take some looking at the batteries with a voltmeter to get a better idea of what is actually happening. For long term stays, it may be better to invest in a seperate battery charger. One that is newer and has all the good features of new electronics that start high and then taper off far better!
Do you know where there is a switch or breaker which would turn the converter off when you don't want it overworking things?
What would be good?
A charger that would charge the batteries at higher voltages like 13.5 when they are down, but taper off when they get near 12 or so and then just barely feed power in as we use power out!
Kind of like blowing up a balloon that has a tiny hole in it? We want to blow faster to fill it but as it gets full, back off to just enough to keep it filled without blowing it up!
Testing voltage takes some idea of what's up, too, as it can confuse us. When the charging is going on we will test and see the charge voltage but the battery itself may not be charged all the way down through it, so we need to come back several hours later after taking the charger off to get an honest idea of the true battery voltage!
I think getting the batteries right and charging better will let a lot of other troubles go away! Not saying there are not other things wrong but this may be a good place to start!
See how that fits and hang around for more questions if needed!