With shore power cord not connected to power, measure the voltage at the battery(s). Connect to shore power with cord and measure voltage again. When unpowered, your battery should be at 12.7-13 V. When connected to shore power, the voltage should be 13.5 V or higher.
Alternatively, you could remove battery and observe if the lights and other 12V devices work. If you still have 12V power, the converter is at least supplying some current.
Another way is to measure voltage directly at the converter's output terminals. When powered, the converter should supply > 13.5V.
For more focused advice, share the year, model number, and brand names. Some converters are known to have issues.