Loud Water Pump

Yup, folks have used foam to mount the pump on if it drives you nuts. Here's one idea.

I guess I left out some info.

I had a full water tank also. The pump did turn off after I turned off the faucets. It has run smoothly before and very quiet.

Where should I be looking for my water pump? It sounds like it's under the shower. Is the pump attached to the trailer frame and somewhere else?

I often run my pump when connected to city water in places (usually on a well) where the water pressures drop so low that I don't have enough pressure to suit me when I am showering. It doesn't hurt the pump at all. As city water pressure drops the pump comes on and will pump from the fresh water tank until the pressure comes back up above the pumps set point.

Noise is often due to the water pipes/tubes vibrating against the structure. Sometimes it can be fixed by mounting the pump on a thick pad and using flexible lines to and from the pump.
Winnebago did all of that but somewhere between the pump and the bathroom faucets the line vibrates and causes more noise than I like. I have just learned to live with it.

Having the city water connected and the pump on is ok, if you have water in the freshwater tank.

This happened to me. We were connected to the city water and no water in the FW tank. The wife turned the pump on while I was out walking the dogs and it ran for a good while as it could not pressurize using water from the FW tank. I came back and heard the pump on and immediately turned it off. A few day's later I was able to add water in the FW tank and turned the pump on. It did take it longer to charge the system but it does work. I'm surprised it didn't burn up.

Easy enough to tell if it burned up. Just put some water in your FW tank and with no city connection, turn on the pump. Does it pressurize the system and does the pump turn off automatically? If you turn a faucet on does it provide water ok?