Where did you source the new climate control unit from - is it a brand new one? Can you confirm that the AC compressor is going on and off, or it could just be the air is being routed through the heater core when it shouldn't be, which perhaps is either a faulty switchover valve or a bad vacuum element.
The default failure mode is defrost and heat I believe, so is it when you accelerate or putting the engine to use that the heat comes on? If so this would point to a vacuum leak/broken heater element.
Otherwise, if it's not this and the system holds vacuum then it may be a problem with the A/C pressure or sensors. Time to do some tests!