OP, I think you may have two problems. Let's start with not getting past 120 psi. If the motor runs fine until we get to 120 psi then the pressure switch shuts things off, that's just a pressure switch that needs to be adjusted. Many of these switches look the same but may be factory preset set to different pressure levels. Not hard to adjust the screws. However, if your problem is the pump keeps running but can't build pressure past 120, that could be a bad valve in the pump itself. A classic reed valve failure is the pressure builds slowly then can't get past some limit. If that is your issue then you need to pull the head off the pump.
The next thing is why can't you restart the pump. That sounds like an issue with the unloader valve not working correctly. Since that valve is typically part of the new pressure switch, I doubt it's broken but maybe it's connected wrong or not at all. Somehow the pressurized air in the pump to tank interconnect is not getting drained after the pump shuts off. Unloader valves are very simple so I think it's unlike to be broken. But it's possible the line is not connected correctly. Do you head a sudden psss of air just after the pump turns off? If no, then it's probably connected wrong. Note that even if the valve never opened, air will naturally leak out of the pump to tank interconnect via the valves in the pump head. However, if might take many minutes to reduce the pressure enough for the pump to restart. This is why you could restart the pump the next day but not while in use. Also, the unloader valve only affects starting the pump. Once the pump start running (just a few seconds) the unloader doesn't matter. I mean if it's open it will take longer to fill up but it will still fill up. If it's closed... well it's supposed to be closed when the pump is running.
BTW, a picture of your system showing the pumping between the pump, pressure switch and tank would be helpful.

(note: in my compressor the unloader line connects right next to the In Tank Check Valve vs at the pump head)