If no air is blowing, it's not the under-hood components of the air conditioning system that is of concern, but rather the blower motor and the controls for that inside the car.
Testing should be focused there. There is a blower motor resistor, blower motor relay, switch and the motor itself, along with wiring and connectors that need to be inspected.
www.repairpal.com/repair-shops
RepairPal Certified shops can diagnose and repair your 2005 Mitsubishi Galant. They provide transparent pricing with a minimum 12-month or 12,000-mile warranty on repairs and services.