I seem to have a similar problem after I dinked around with the pressure screw. Not near as bad as yours, but goes pretty low pressure before the pump starts, and then too high for my liking before it shuts off.
I suspect it has to do with a balance between the pressure tank pressure and the pump settings. I need to craw back in there and start tweaking these again.
Need some good help from the crew which is sometimes hard to get.
JimS on the Delta