These are the exact symptoms of a bad converter. Here's a simple troubleshooting procedure:
1. Disconnect from AC Shore power or shut your generator off.
2. Disconnect the negative battery terminal.
3. Open the converter housing and pull the converter out.
4. There should be one or two fuses near the converter 12VDC side. Make sure they are good by checking them with your multi-meter. If you don't know how, let me know and I can give you instructions.
5. If the fuses are good, carefully reconnect to shore power and check the voltage at the 12VDC connection point on the converter with your meter (where the large wires are coming out). If your meter reads approximately zero volts, your converter is bad. Disconnect from the AC source and replace the converter.
If you have 13-14 VDC on the converter, then there is something else wrong in the circuit such as a bad fuse, a battery disconnect switch in the wrong position, a loose wire, bad battery, etc.