With a Dometic Absorption Cooling Unit (Fridge) that can run on either Propane or Electric, it SHOULD switch to Propane when not plugged in to 120vac. More on outlets later.
NOW: if there is something wrong with the propane side of it, it won't switch (it will fault out)
I can not say more about that specific model... I don't have the bandwidth to research today.
IF the interior light on the Fridge works, when not on shore power.. IT should work on propane, (That is the easy test, more complex tests are dependent on the model).
Outlets: Normally they do not work when not plugged in. YOU CAN add an inverter to power them but that's the only way.
OH, and if you try to run it on an inverter.. What someone else said about eating batteries. BIG TIME.
What make and model converter do you have. that too may make a difference.