Engine idle changes when I turn on the A/C in my 2004 TrailBlazer

Question edited for clarity.
Question moved to model category.

If you get a 4 foot length of 1\4" clear plastic hose, you can put one end in you ear and use it as a stethoscope to track down the humming. The closer you get, the louder the noise will become.

Coolant Flush or engine oil flush?

https://www.google.com/search?q=2004+Chevy+trailblazer+coolant+flush

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Check the idle air control device on the intake

I would suspect the clutch on the air conditioning unit, also if the unit is overcharged
from a recent charge this would also bog the engine. At speed it would not be as pronounced.
Hope this helps.

Sometimes the Computer [PCM] that controls the engine will loose it's memory on how you've driven it when power is removed. It should reset itself after a few miles of driving.

If it has not pulled any codes (idiot light). Clean your throttle body, and change your spark plugs with A/C Delcos #41-103, most other brands will cause rough idle like Autolites and Bosch. This would be a cheap way to fix this problem.

If by chance you have just recently changed the battery or had the battery unhooked. Let me know, it's an easier fix that costs virtually nothing.

Induction Service: hose is loose. Just had this fixed thru a dealership.

Assuming you mean the engine dies, you need to clean the throttle body first of all. There is a build up of grime in the throttle body and on the throttle plate preventing the plate from closing completely at idle.

Full engine tune up , with fuel injection cleaning.

Assuming you have the correct size battery, this really shouldn't happen! I am assuming the engine is idling at fairly low RPMs, so when the AC kicks on, it kills the engine? It might be that you need to drive the car for a bit so that the engine computer can "relearn" your settings, which disappear when you change the battery. Or, when your battery died, one of your sensors might have gone with it. There is a idle sensor on most cars that raises the idle RPMs when the AC comes on.

try changing the air filter ,i had a problem ,at ilde in drive my trailblazer would stall, changed the air filter now it's not stalling and got my power back ,there is a secondary air pump that i have to change ,
not sure were it is ? still not running perfect but it's running better .and my service light is on too.
i got it put on scope ,it's the pump .
not sure if this helps ,but i'm start to run in to problems too,i have a 2004 trailblazer.
i like to do fix things myself ,i hope i can.it's the cost if you know what i mean.