When the Dometic refrigerator operates on AC or DC, It uses heating elements (different one for 12 and 110 volts) to heat the ammonia in the refrigerator. When operating the refrigerator using 12 volts, this heating element operates at about 12.5 amps. That is a good amount of current. If the battery is not being charged, it is going to drain pretty fast. Also the voltage drop from the wires, going to the refrigerator, is enough to lower the voltage at the refrigerator a fair amount, which means lower heating temperatures at the refrigerator (could be bad in cold temperatures). If you have a volt meter, you can measure the DC voltage at the refrigerator when it is operating on DC with the truck running and not running to get an idea of what kind of voltage drop you are experiencing in your setup.
I replaced the wire between the battery and the refrigerator with a larger wire (I think 8 gauge). I added a fuse at the battery and bypassed the fuse box, which can also be a source of voltage drop. I think the newer fuse boxes that FWC use are an improvement over the older ones. My refrigerator seems to work very well when I operate it on DC while driving. When I first got the camper and upgraded the wiring, I almost always ran the refrigerator on DC while I was driving but every once in a while I would forget to switch back to propane at my destination. The worst case was maybe an hour after I arrived but one of these days, I will forget for a longer period of time. Remember, each time you run the battery down, you are wearing it out. I realize that the amount of propane I am saving doing this is pretty small and the newness has worn off so, unless I have a long drive planned for the day, I just drive on propane.
My Dometic refrigerator turns off when the temperature in the refrigerator reaches the target in all three modes of operation. It does not stay on.
Another thing that is worth being aware of is that the auto reset breakers between your truck battery and camper battery, at least in my camper, are 30 amp breakers. If the batteries in the camper get low, your alternator in your truck is able to deliver more than 30 amps to the camper batteries (I have measured 60 amps once in my truck). If the amps gets above 30 amps, the (in my truck there are two) breaker(s) will start tripping and resetting. This is something that is kind of hard to know is happening. The camper batteries will eventually get charged enough that the cycling of the breaker will stop. If your 3 way refrigerator is on DC, this breaker cycling (if it is occurring) could go on for a longer period of time. I don't know how much longer.
Like others on this forum, I replaced the factory wires between the truck battery and camper battery with larger wires, and replaced the breakers with higher current ones. There are laws that FWC have to meet and the setup that they use meet these regulations. Mine probably does not.
So, to play it safe, it is probably best to stick with the gas position while driving with your camper unless you want to experiment to better understand the personality of your configuration.
Steve