We keep our fridge on 24x7 (parked at home) on electric. But it's set on "Auto" so it will flip over to gas if there's a power outage. We leave one propane tank on also.
When traveling, we don't do anything except load the food the night before. Anything frozen, we attempt to put it in the main freezer in the house and get it frozen first, then put it in the camper the evening before.
Traveling, nothing changes. The refer runs on propane and when we reach our site it automatically switches back to electricity when we plug the camper in.
You're 3 way should switch to battery only when there is no electric and no propane.
Just flip the switch on "auto" and have fun!
Good luck. Congrats on your new camper!