If you're dry camping, fridge will stop working. Many have a "Check" light on the top. If it's lit, you are out of propane.
Did you mention what kind of tanks you have? If portable tanks, you can pull one and take it to town to get it filled.
If a MH, Extend-a-Stay (mod) will allow you to use a portable tank instead of the inbuilt tank.
As far as anything being damaged by running out? Each device should try to light a few times, then lock out (stop trying) when you run out. When you do fill up, hold a BBQ lighter against the stove top burners and open it full until it lights to purge the air from the lines before power cycling your other propane devices (fridge, h/w, furnace)
Hope this helps.