That's good to hear. Make sure you check the grounds too. Those connect to the frame and often are exposed to road water and salts so can tend to rust and corrode along with the terminals right on the batteries.
Remember to take a picture or make a drawing of the battery wiring if you're going to continue with the battery maintenance at this point (suggested).
But all that battery stuff wouldn't have anything to do with the 120 Volt circuits (shore power). The shore power comes into the house at the transfer switch, through the Normally Closed contacts of that big relay and routes to the Circuit Breaker panel. From there to all the house circuits. No battery power involved.Take all the batteries out of the rig, and it'll still have TV.
So keep us posted about what you find with the shore power issue.