There is a glue / tar like substance that is applied when the vehicle is new to the blend flow door inside of the heater box.
The glue, when it gets hot, tends to bond the door to the opening.
As long as you leave the AC in the cold position - all the way, it should keep blowing cold air. At some point, the evaporator will ice up and air will not flow through it and although it will look like a solid block of ice, you will not get any cold air.
The solution is to shut off the air conditioning and leave the controls in the recirculate position.. You will probably have cold air coming out of the heater vents for 60 - 100 miles before it will defrost.
There is a tool used to replace the blend flow door actuator, the job is best left to the dealer - you can screw up more then you will probably fix.
Putting heavier rads and coolers will not do anything to fix the blend flow door problem.