To test DC side of the converter, look at your AC/DC distribution center. There should be the DC + and - terminals available there somewhere! When plugged into shore power, your converter should be putting out something like 13.2-13.6V or possibly more if it is a multi-stage smart charger. If it is something like 12.7V or less, then the converter isn't outputting DC. You should also check the circuit breaker near the positive battery terminal, it looks a bit like this.
It may not be resetting properly and needs to be replaced.