The shore power is relying on the converter AND the wires from the converter to the battery to do the job. Many many converters have been replaced, and of course when that happens the wire connectors are also tightened so it looks like the new converter fixed the problem.
Before you do anything, measure the converter output right at the converter. My lights were dimming, and I found my problem was just a loose negative wire at the converter terminal going to the battery. Tightened it up, and all worked just fine again.