A few things.
1. Switches are cheap and relatively easy to replace. It is the switch that controls pressure, so you are looking in the right place.
You can try turning off the system and filing the points with an emory board, but that is probably not the source of the problem.
2. There is a little pipe leading to the switch. Sometimes the pipe gets so clogged with crud that it makes the switch sluggish in turning off/on. Check out the little pipe.
3. You can, of course, adjust the switch, and that is the
first thing I would try. The site below gives a great description of how its done. Still, this "phantom" increase in pressure is not a good sign for your switch, so I'd sure keep an eye on it.
How to adjust water pump pressure, pump cut-on pressure and pump cut-off pressure - private pump and well system do-it-yourself repairs