We had a 2001 Winnebago Adventurer and experienced a similar problem. I believe as others have said you probably have a problem with the city water/tank fill valve. When we first started RVing I filled the tank by switching the valve to the tank fill position. I shut off the water at the source and disconnected the hose. Every time we tried to use the pump it ran continuously and had very low water pressure. I discovered I'd left the valve in the tank fill position. The pump was recirculating the majority of the water back o the tank. A simple flip of the switch and everything was back to normal.
So to your question are the 2 connected? The answer is yes. To the other question do they ever fail? The answer is also yes. But, before you change it out disconnect the hose and look into the connector on the motorhome. There's a spring loaded valve that's supposed to seal the hose connection when the hose is disconnected. If it gets off center it acts similar to leaving the valve in the tank fill position. When using the pump water will recirculate to the tank.
If this is the problem you may be able to correct it by removing the hose and depressing and releasing the spring loaded plunger a few times. If the seal recenters itself you should be able use the pump normally with the valve in the city fill position. If this doesn't correct the problem it's probably time for a new valve.