Don't run it when it's empty. Any fridge, whether electric or propane, runs better when full. It's more efficient cooling liquids and solids than it is cooling air. You should only leave enough air space in it to allow the cold air to circulate between the other items. If you don't have enough stuff to keep it full, add bottled water or something to fill the void spaces.
But don't put anything warm in the fridge, either. Pre-chill items in your home fridge before putting them in the trailer fridge.
And just to monitor the freezer portion, put Dixie cup of ice in the freezer, with a penny on top of the ice. As long as the penny remains on top of the ice, you know the freezer hasn't thawed. If you ever find the penny INSIDE or UNDER the ice, you know that you've had a thaw and you'll have to throw out all of the meats and other perishable items that can't safely be re-frozen.
Engineering: Finding complex solutions to simple problems you didn't even know you had.