Notsure the F14 advice is correct, for your coach.
It IS, in relationship to the current models of inverters that are being used, but if I'm not mistaken in your model and age of coach you just have two small inverters.
Therefore I think your TV's will indeed just always run off the inverter. Your converter (charger) will take the shorepower 120vac and charge the house batteries. Your inverter will take the 12vDC and invert it back to 120vac. That was a common cost cutting method in a number of coach manufacturers who weren't putting in the $1500 automatic converter/inverter/transfer switch devices.
DaveB, Raleigh, NC
2015 Tiffin RED 33AA, w/Honda CRV
VMSpc, Magnum BMK/ARC50