Yes, 120psi is too high. You would run the risk of having the water heater relief valve open, plus it would be hard on the other plumbing.
Are you on a well or is this city supplied water? If it is city supplied water, they are responsible for installing a pressure reducing valve to control your pressure. Ideally, you should have about 70psi. The cost of a pressure reducing valve is about $60 to $80.
If this a well system, you may have a faulty cut off switch for the pump to your holding tank. I would suggest you have a well pump installer inspect the installation and recommend repairs.
Good Luck!
Dodgeman