All the TrailBlazer/Envoy offerings used the 4L60-E model transmission - except for the SS option TrailBlazer (4L70-E) ...
The 'Solenoid A' is the low/1st mode activation for the trans - this signals the trans to engage low/1st when the TCM grounds this circuit.
The matrix for the two (A & B) solenoids is as follows;
Solenoid activated = Gear
A = 1st
A&B = 2nd
B = 3rd
None = 4th/Overdrive
The solenoids are supplied 12vdc by their feed wiring and then engaged by the TCM via grounding of the units - this ground signal is pulsed by the internal circuit to modulate the shift quality dependent on various data inputs from the ECM of engine load/dynamic.
If there is a problem you will usually feel a noticeable difference in the trans shift modes and qualities while driving.
So either your 'A' solenoid is
a.) stuck or there is dirt/debris in pintle/orifice
b.) not getting a modulated ground signal from TCM due to wiring or ???
c.) the coil has failed
The coils are pretty inexpensive, compared to other trans parts
, and aren't to hard to replace - BUT ... be sure it's actually a failed solenoid causing the issue or you're wasting your time.
If the trans doesn't have a good service and maintenance history I would suggest a fluid & filter change before spending a lot of time and money chasing other things ...
While you're in there for the service you can check solenoid connections and for 12vdc feed - BUT, do not attempt to ground or activate the solenoids without a dummy load or you will take out the TCM!
Hope some of this helps.
ps: I have a 2003 TrailBlazer EXT with the 4.2/4L60-E and love that vehicle.
Approaching 200K and LOTS of hard use (towing, offroad, just fun bashing ...) and it is still hanging in there