First off, welcome to AirForums!
Could it be a faucet washer is "old" or a mesh filter has debris blocking it? Water could be running down a pipe under the cabinet...
Does it also leak during the day if you use the toilet or sink? Or just first thing in the morning? Has it leaked at other campgrounds or just this one?
One thing you should do (instead of guessing) is get a water pressure regulator with a gauge
to hook-up at the faucet of the campground next time...peace of mind. We use the one w/o the gauge (less out of pocket if you forget it at a campground!)
and then have a permanent gauge installed inside our trailer we added when we re-plumbed the whole thing during our restoration.