First thing that comes to mind is "Is your pump capable of producing 60 pounds of pressure?".
Do you know the characteristics of your pump and well?
Assuming it is okay, you need to do two things, adjust the pressure in your bladder tank to the new, higher value (in this case 38#) which is done with the pump off and a water valve opened to drain off any water under pressure, and then replace the switch (see next, though).
The new switch should not need adjustment...in fact, your old one could be adjusted to raise the pressure - it doesn't need to be replaced. With that in mind, you could save the new one as a backup if the old one dies. The larger adjustment changes both the on and off settings while maintaining the separation (i.e., 20 pounds). The smaller adjustment changes the offset between the on/off settings.