You haven't mentioned a converter problem before now so don't know what you're referring to...did you mean refer problem?
Yea, the refer does need both 12V and 120 to run on shore power and if your converter isn't charging the batteries, it would tend to work erratically.
Thing to do is find out if the converter is working, just take a digital meter and measure the voltage across the house and chassis batteries. Should be 13V+. Many older rigs don't have a provision for changing the chassis battery while on shore power though and most people add a float charger to do that. Around $20.
You might find an Aux Start or Emerg Start switch on the dash. That jumps the house and chassis batteries for starting up the rig, but first measure the voltage and see if the converter is working. If it isn't you'll need a jump.
BTW, converters can be on GFI's circuits so make sure that it's getting power.