When you are connected to an external water supply, just turn the water pump off. There are built-in valves that control the water flow. Also, FYI, water will not back-fill into the fresh water storage tank. You still have to fill that tank manually.
Not quite sure about your statement about the refrigerator. However, if you are running on 12-volts DC (batteries), you need to make sure the propane is turned on. Also, if your Airstream has been in storage for longer than a day or two, you may have to purge the air from the propane lines. This is most easily done by turning on the propane appliance the farthest away from the tanks (in our 19-foot Bambi, that's the hot water heater).
That device may also take several attempts to start, while the air is being purged. For example, the hot water heater may attempt to start, then shut down. You'll need to turn it OFF and then back ON again, perhaps several times, until it starts and stays on.
After that appliance starts and runs on propane, your refrigerator should start automatically. Not sure about your model of refrigerator, but some later models have a couple of switches on the front panel. Make sure the settings are correct for using propane. If your refrigerator burner doesn't stay lit, it will shutdown after a couple of attempts. You may have to turn the switches OFF and back ON again to reset the ignition cycle. (Have someone stand outside and listen for the click-click-click of the electronic igniter, and the slight whoosh of the burner lighting).
If you are connected to 110-volts AC (shore power), your refrigerator should automatically switch over to the electric resistance heater. However, again, you may wish to verify that the switches on the front panel are properly selected. Most likely, AUTO will be the correct setting for your refrigerator, if that is one of the selections.
When you get time, you may wish to search this forum for additional information on installing a quieter, external fan and on using a small battery operated fan inside the refrigerator to circulate the air and help it cool more evenly.