Sounds like it didn't fill the tank. When you said the sound of water filling stopped and the sensors didn't change, what did they read? Did they read empty the whole time, or did they read full?
There will be an overflow for the tank, they all have one or we'd be reading about ruptured fresh tanks all the time. Don't turn the hose on too hard so the overflow can keep up and not let the tank over pressurize. I usually turn my hose on just a little bit and let it slowly fill the tank. It take a while (maybe 15 minutes to fill), but I'm doing other things at the same time and don't want to cause any damage if I forget to turn it off.
I wouldn't worry about it, RV water pumps can run dry without any damage. Yes if you have water in the tank the pump can run for a few minutes while it fills the system including the water heater. Pump flow rates vary between 2 - 4 gallons per minute. So if you have a 10 gallon water heater it might run for a while.
I would hook up the hose and use city water to fill the water heater and bleed the air out of all the faucets. Then flip over to fill the fresh tank and wait until the water starts coming out of the overflow.