Unhook your battery. Secure your battery terminals so they do not touch metal.
Hook up to shore power. Turn on several 12 volt lights. If the lights do not come on or are not at full brightness, you need a converter. Putting in a new battery only tells you that that battery currently has a good charge. It does not tell you if the converter is doing it's job.
I bought a new battery today because the dealer confirmed it was dead. I performed the above test and unlike a previous use of shore power without battery, this time I had nothing. Looks like it is a multi problem of converter and battery. So I guess everyone was right in one way or another.