The sw can be replaced , but you loose the freon ..
However, the sw is a pressure activated trip and it is more likely that your a/c system is short of freon. This results in the high side pressure never getting to the sw cut-in spec....
Look at the drier sight glass for freon charge level before changing the sw...or test high side pressure w/gauges..
You have fan when jumping sw. so you know the electicals of the fan circuit are fine [ with the exception of the sw circuit]