So grateful for all the advice! We're still trying to figure this out and appreciating any insight. So a voltage meter would be a smart thing to get and keep with us to check any future sites to make sure that the output is what it ought to be, right?
We're in touch with the couple who sold it to us, and we're under the impression that they never had any of these problems. However, they could be: not telling the truth (though that seems unlikely, as they've been very helpful in the past with teaching us how to use it, even talking to us on the phone when we were camping and trying to help us figure it out); not experienced with more rustic sites (we were in an electric site at a national forest campground, and I think they tended toward RV parks when using it); not understanding what we're describing since we're maybe not always explaining it the best. For example, we also had trouble filling the water tank and thought we had damaged something since water was coming out the bottom. Turns out the valve was open but no one had even told us that a valve existed! So we're learning as we go.
So the advice so far is: Check the voltage of a campsite, and use no extension cord when possible, a high powered one when necessary.
We didn't have the water pump or water heater on at all (due to not being able to fill the tank!) but used TVs (plugged into AV outlets), refrigerator (usually on AC, though had it on gas for a while when we thought that might fix the problem), sometimes a fan (plugged into AC outlets), once A/C for a couple of hours early in the trip (I think that uses DC?), and the lights, which were always the canary in the coal mine that told us when we were out of power. Is there an easy way to check the converter and tell whether it's the problem? We're also going to try to talk again with the couple we bought it from and see if they can help us out!