My rear AC and heat blower stays running after I

you see those fuses, pull/replace them one at time till you pull the fuse that kills the blower motor, then we go from there. see the 40 amp fuse on the left, that is for hi speed. The two middle fuses should only go hot with the key on. Everything off and key out use a test light to check voltage at those two fuses, shouldn't be hot? I can only help with testing. I'd hate to tell you to replace something and it not fix it.

2002 - 2009 GM BLOWER MOTOR RESISTOR - YouTube

? 4:17
https://www.youtube.com/watch?v=WWi-_-kY-dM
Sep 5, 2012 - Uploaded by davesgarage512
I have a 2003 GMC Envoy and my issue is when I shut the blower off it still ... Can this issue also be remedied ...

2004 GMC ENVOY Blower Motor Control Module / Resistor

The relay in your blower motor is stuck on, there is a TSB for this issue. Replace the blower motor resistor and you'll be back in business.
How does the resistor get power if the fuse is pulled?
Replacing the resistor will cure both problems. TSB is a technical service bullitin seen below
Subject:HVAC Blower Motor May Run Continuously After Vehicle is Turned Off (Replace Blower Resistor Relay)Models:2004 Chevrolet Avalanche, Silverado, Suburban, Tahoe, TrailBlazer, TrailBlazer EXT2004 GMC Envoy, Envoy XL, Sierra, Yukon, Yukon XLwith Manual Control HVAC (RPO CJ3)Built Prior to August, 2003

you may need a new Blower Motor Control Module. get it from the dealer or AC Delco parts house.

the rear blower module is bad. when you take the rear blower cover off, you will see it wired to the blower motor.

will try to help but i think you'll need to have that blower replaced. if it's coming on for a sort while and then shuts off after running a short while it sounds to me like the blower motors heating up inside and then shutting itself off due to a short in the blower motor itself. no really quick fix i know of other than replacement of it. hope this helps and thank you for choosing fixya.com

does it work on heat mode if not then it is probably the blower motor switch in the dash or the blower itself

you need to check / replace the blower speed resistor module that should take care of the problem it is located lower evaporator case right hand side very bottom by the blower motor.

Check your ignition switch by checking the voltage at the fuses for affected circuts vs. other circuts that are working well...

do you have power to it..or a bad ground around blower motor? is the head unit not signaling the blower to run?...calling it with A/C or heat or just fresh air ..try all settings and advise...good luck