Here's a thread on troubleshooting my 12V converter.
I may be confused, but I only thought you had 120V when plugged into shore power.
When running only off the battery, I thought only 12V items were powered.
Check the terminals on your converter output plugged into 120V and battery disconnected.
You should be getting about 13V. If not, the converter is bad.
Read my thread, it has a lot of pictures and where to check your converter.
As far as not having 120V outlets plugged into shore power, that's a different problem.
Check for a GFCI outlet somewhere and see if it trips/resets.
Even if your 12V converter is bad, you should have 120V outlets when plugged into shore power.
May also check voltage at outlet you are plugged into shore power and make sure the breaker isn't tripped on that circuit.
Let us know what you find.