Sorry to hear of your trouble. Here are some things you can try.
1. The pump can run for quite a while without water so short trials won't harm it especially of you give it a minute in between tries.
2. The city inlet has a 45 psi regulator so you won't have much luck getting water into it unless you hook up to a municipal water source.
3. Your manual will describe how to de-winterize. It is your friend, you should get it out and follow it. They are also available online. https://www.airstream.com/owners/manuals/
4. You need to fill your fresh water tank with a bit of water. The pump takes its water from your fresh tank.
5. City water inlet bypasses tank and pump, goes straight to cold water lines so it won't work for you as mentioned before.
6. Other inlet is most likely the sewer tank clean out and that is why it is draining out. Close the sewer valves when not draining to avoid build up in the tank. See instruction manual on that.
7. Generally behind a small locked access panel is the fill inlet for the fresh water tank find it and fill the tank from there. See manual for location.
So get some water in your tank so the level indicator at the pump switch is showing you have at least some water in the tank. Then the pump will fill the hot water heater and deliver hot and cold water.