The inverter should have a small buzz coming from it if it has a load on it. It will from time to time go silent if the batteries are fully charged and there is no draw. This is normal.
When the inverter buzzes louder, there is a substantial draw on the 110 side inside the coach (i.e. using your house vacuum cleaner, doing modifications and having a drill, halogen work light etc. plugged into outlets...like me most of the time...
). This too is normal.
There are two breakers (white push button types) on the front of the inverter near the on/off switch. These should be pressed IN for everything to work correctly. If they "pop" out, you may have overloaded a circuit temporarily or something "could" be amiss in a device or wiring.
99.99% of the time, electrical issues (whether 12V or 110V) in a coach are related to a "bad ground" somewhere. Devices and components do fail but not that often...unless the mfr. got a bad batch or installed them incorrectly.
Here is another thing you should change since the PO might not have done it when winterizing it. Change out the water heater anode and pressure relief valve. Both are about $15 each and are easy to change. Both parts are usually available from RV parts houses or dealers and possibly home improvement stores. You should also vacuum out all areas of your coach interior heaters and refrigerator compartment, check the coils for damage and debris on your roof top AC units, as well as the condition of the shrouds.
There is a host of items you should check re: the coach/chassis so send me a PM with your personal email address and I'll forward the info to you for your review.
BTW, I am glad to help and it is funny that you are now the third person to purchase virtually the exact coach I have. I thought I was alone out there with the specifics so the least I can do is share the info I have optained.