Couple of details
1. When connected to shore power my 110v AC outlets even when verifying that the inverter is off (so i'm getting shore power)
huh?
2)
This is probably just the starter battery voltage. 10-11 is low, so you may want to check the connections. If the key switch connects the house & starter batteries together then you may want to make some changes. You normally don't want the house connected until the alternator is running.
You need to figure that out. Its way more fun to work that out ( and create a schematic) in your driveway than it is on the side of the road in a mud hole in a freezing rain while laying in an anthill and it snowing, and your SO is yelling at you and your friends are waiting on you.
3)
Typically, a solenoid ( ACR) of some sort connects the two batteries when the alternator is running. That kind of button would allow you to "jump start" your rig from the house battery if the starter battery got low.
4)
My suggestions
Find the batteries. Check the voltage on the house battery when its plugged into shore power. If its less than 13.2 then you have an issue. Ideally the convertor would have a 3 stage charger that would run as high as 14+ volts, but drop down the voltage once the battery is charged. You might also try checking the voltage on the house battery with the headlights on . If the house battery voltage drops when you turn on the headlights, then they are all connected together all the time. This is not a good system.
Let us know what you find.
Tom