Water run-on noise after faucet is closed

Is it the check valve on the output of the water heater?
Sometimes they're noisy.

If it sounds like the water is running for a few seconds and your on internal water,,,,, it is the pump bringing the system back to full pressure.....

If you are on (city) water, do you have a pressure regulator on your incoming line next to the RV, if you do put the regulator close to the city water valve. Sounds like a vibration as it comes back to the 40 lbs on the down side of the regulator.

I have a cheap inline regulator, right at the CG spigot.

Try a new regulator.

Try it without the regulator for a short time before you waste money on a new one.

Sounds like the lines just getting back up to pressure

Many Hitchhikers had accumulator tanks which could cause the sound you are experiencing. Also, I had to replace the city water inlet valve and the new one makes a slight sound when water is running.

I had the same issue. It was the city water inlet that was causing the noise. Changed it out and everything is quiet now.

If you are connected to city water, your on-board pump should never need to run.

I'm assuming you are talking about running your on-board pump and water in your fresh water holding tank... yes?

If so, you have air in the line somewhere. Mine does this sometimes, especially after draining the water heater and tanks when returning home from a trip and then filling everything up again for the next trip and pressurizing the water system again.

I have found the shower line is my biggest culprit when this happens. Easy fix is to simply turn the water on at the shower and let it finish spitting air.

One more thing that can act like an accumulator tank is your water heater. If you have recently drained your water heater, when you fill it again, there is an air pocket at the top of the tank. This air can make the system act like an accumulator tank.

Still, it's not uncommon for the pump the run another couple seconds after turning off the faucets. Now, if the pump continues to run for 30 seconds, then you've got some serious air in the line somewhere, or else your pump needs to have the high and low pressure on-and-off re-adjusted.