Is the pump above ground, in the pump house? If so, you have a jet pump and the problem may be that the tubing to/from the pressure switch, the tubing fittings, and where the tubing fittings screw into the wet end of the pump and the switch, may be blocked with rust. That prevents the switch from reading real time pressure, and it lags at both start up and shut off.
You must unplug or otherwise shut off the pump, drain the pressure tank and remove that tubing its fittings etc. and clean/open all of them. You do that because you have water pressure on the tubing etc. unless you drain the tank.