First, if you haven't done it already, you need to invest in a Progressive Industries EMS HW-50C unit.
You can Google or Bing for some really good prices but the lowest I have seen them is on RVupgrades.com for $333.
This unit will CONSTANTLY evaluate the quality of your shore power and with shut off power to your coach and appliances when the power exceeds the normal operating specifications for low and high voltage, low and high Hz, loss of ground or neutral. It also has 3580 Joules of Surge protection.
I think most of your problems are most likely coming from the shore power supply. Especially since all is well on generator power but currently you have no way of knowing that specifically.
If you had the HW-50C, it would give you codes to tell you exactly what is wrong with your incoming power supply.
BTW, you had posted this in a very inconspicuous location to get any useful information. In the future, scroll through the categories and choose one that best fits the problem that you are having and submit your post there.
I came across this SOLELY and by accident when clicking on "unanswered post". It doesn't belong in the category - Forum Issues.
Just a heads up!
Dr4Film ----- Richard