I know you said it's filling your tank with city water fill, with that said, there is usually a check valve on the water heater outlet side. mine was on the heater unit itself. If that check valve is letting water through it will fill the water heater tank as well as the fresh water tank. Mine went bad and did exactly what you are describing. If the water pump has seen a better day I would replace that as well. Both parts are dirt cheap and easy to replace. Easy check by taking out your water heater plug and see if water flows out as you turn on the city water. That should fix it. The pervious owner might also have put in a diverter valve.
Good luck, Joe
PS This is all assuming you have the water system on normal and not winterize. There are three winterizing valves in my 1181.