I had the same experience when I got the new style pump as a replacement. Very frustrating, I adjusted it using the directions supplied by Old Biscuit and adjusted the screws as much as five turns from the factory settings.
Every time we would go out it would require adjusting again, I even had pressure gauges in line to verify my adjustments.
I was unable to solve the problem until I installed a one gallon accumulator tank.
I then adjusted the pressure side to just 50 pounds to help conserve water when dry camping. I always wanted to try an accumulator tank and I like the way it works, the pump runs on demand and fills the tank and does not run again until it is under demand again.
I still like the old pump better, it was a variable speed and had no short cycling, and certainly would not run with the faucet closed!