I'd make a diagram of the wires in case they fall out of the plug.

Old plastic can stick together making things difficult to disassemble. In this case, rocking the plug around can free things up rather than a straight pull.

This looks like the old round pin type connector, I'd be tempted to remove the plugs back cover ( how ever that comes off. ) , using a sharp pointed tool to lever the pin sockets out then working the plastic housing out of the switch.

Got a pic of the new switch?