Your Trailblazer's air conditioner is working just fine, but when you flip the switch to the dash vents, it sounds like the air is blowing up against a wall in the dash and no air is coming out. The floor and defrost positions both work, but on a hot day you need that cool air flowing from the dash. The switch that controls this is called the mode actuator and its job is to move the little door inside the vent tube to direct the air flow to the dash. Replacing it can be done with just a few tools and requires little mechanical know-how.
Locate the screws holding the dash panel below the steering wheel on the driver's side. The two screws at the top may have plastic covers over them and you need to pry these off carefully with the flat screwdriver to expose the screws. Use the 9/32-inch socket to loosen the screws. Once you have all the screws out, the panel should come off. Put the screws in a safe place so you don't lose them.
Look up under the dash to the left of the console and locate the grey vent tube coming out of the blower unit. Pull the tube out from the unit and, with the long, flat screwdriver, pry it loose from the firewall mount just to the left of the steering wheel shaft. It is held on by a plastic grommet.
Locate the two mode actuator boxes next to the blower outlet. The top one nearest the windshield is for defrost and the bottom one controls the floor and dash. Take out the two screws of the bottom one using the 7/32-inch socket driver and pull the wire connector off. Be sure you note the position of the plastic white gear sticking out of the switch. This is what makes the vent door move from one position to the other.
Align the plastic gear of the new switch to match the position of the old actuator. If you have lined it up correctly, it will drop right into place. Reconnect the wire and screw the new actuator back onto the unit. Replace the vent tube to the blower unit and snap the grommet back onto its mount.
Put the dash panel back into place and replace the screws. Now test your repair work by turning the ignition key on, switching on the AC and turning the selector to the dash position.