It indeed could be the inverter at fault, but that is far from the only possibility. The 30A breaker goes to the inverter unit. The inverter has a transfer switch (a relay, basically) that normally will pass through the shore power when it is on to the connected circuits, or switch to send them inverter power when needed. It sounds as though this may be switching back and forth when by appearances it oughtn't be doing so.
I would check carefully for a loose connection somewhere between the 30A breaker and the inverter to start with. This could be in either the neutral or hot line. I would also monitor the AC voltage on the two legs; if one is high and the other low (possibly alternating or switching somewhat randomly) you almost certainly have a poor or loose neutral connection somewhere between the power company and your RV electric panel. If that is the case get it fixed post haste.
