My air is not blowing out of the top vents in my 2006 Trailblazer and the water from the air conditioner is running back inside on the floor on the passenger side

These are run by o relay blocks that are located behind the glove box. These control the door servos' that direct the air flow. Try tapping on the with a screw driver handle. Some times they are stuck and not connecting. Hope it helps

verify the clutch operation of the AC comopressor first

this might help
http://www.fixya.com/cars/r6571076-blower_motor_not_work

I’m happy to assist further over the phone at https://www.6ya.com/expert/jeremy_d728a59f986299fa

will i have bad news for you, your heater/a/c system is all electronic. ( not like the old days when we seen this there was just a broken vacuum hose ) now all the doors inside the heater system are run by electric motors they call them actuator's now days and they are all controlled by a computer called a HVAC control module, and most likely this is what has went bad on your truck.....sorry

flapper valve not closing properly..usually vacuum controlled..do you hear flapping opening and closing..this is inside the duct work

Hi. Have your car aircon evaporator cleaned. Check also the drain hose. All water should go out of the car through the drain hose. There might be something clogging up the hose that is why water will run out the floor and that causes also the aircon to work intermittently.

Have a good one!

under car pass side just behind tire on firewall looking for tube that makes 90 degree bend make sure not clogged if still leaking check for cracks in the evap case

I would start by checking all your fuses. Then I would check the A/c control panel on your dash. There is little vaves inside those black boxes under the dash. One side is vent the other side is a/c. it might be stuck. knock on it with something blunt.

thereis an actuator controling the flap t may be cracked causing the problem

Most likely a clogged drain for the condenser. this is a common problem.