that knob on the right should tuen it on and off too. havent seem many problems like this on here. you might want to get some help, auto electric shop to trouble shoot it. the control setup and the comp would be pricy stuff.
theres a ton a pinpoint ohm tests for the entire system. theres a list of things for no heat, and a different list of things for no air, but sounds like your comp is the issue. you can check the voltage to the relay switch.
05 V6 auto,
"The V6inator" torch red, base, amustangrocksleather padded leather armrest, shift boot and e-brake cover