Question about house battery's

Hello everyone, I had the camper out for the first trip of the season and ran into a few electrical issues. First many outlets did not work, I found that a breaker was tripped but tripped instantly if flipped back on. Since it had 2 hot wires going into it, I tried each one separately and found that 1 caused it and the other had no problems. After this everything worked, I am just not sure what that wire was feeding since I cannot find anything off now and it was marked "general purpose:. The issue I still have and possibly unrelated was low power. Even when I am plugged into shore power, if my furnace kicked on, I would get a Lowe power alarm on the refrigerator and my lights would be extremely dim. Even if I switched to LP for the refrigerator it would not work unless I started the motorhome. I am hoping it is just a bad house battery, but would that have this effect even when plugged in to shore power? I had it tested at a parts store on the way home and it did say replace ( had 330 CCA rated for 600). The battery is 4 years old. Any input would be greatly appreciated