I just wanted to toss this out there for anyone that might be chasing a charging issue.
In our family's automobile fleet, we have cars that either do not have a volt gauge in the cluster, or at times, the existing volt gauge reading might be suspect.
I found this little gizmo to be very handy in providing a look at the state of the system voltage and charging level, without having to rely upon a missing or suspect cluster gauge.
INNOVA 3721 Battery and Charging System Monitor
I ended up just buying one for each of our vehicles and keeping one of the devices in each console.
You can use it to check the non-running state of the vehicle's voltage as well as the charging rate of the running vehicle.
Since you believe you may have a current drain that is resulting in a large voltage drop overnight, I would not leave this device plugged in overnight, but instead, plug it in before you attempt to start the truck in the morning.
Your truck, being a first generation Avalanche, should be showing a system voltage of around 14 volts or so while running.
Especially right after you start the engine.
A much lower voltage usually indicates an alternator that is not charging properly.
This does not apply to the second generation trucks, since the charging systems in those trucks is designed to fluctuate with demand in an effort to extend the life of that system and the battery, as well as, reduce drag on the engine due to constant charging by the alternator.
It is important to note, even if your battery shows a good voltage level at rest, if there are dead or dying cells within that battery, it will not actually have enough power stored to perform heavy tasks, like running the engine's starter.
Light weight power users. like lights and radios, may appear to be functioning normally, but they do not need as much amperage to work.
I think I would start by having a different auto parts store perform a LOAD TEST on that battery, just to make sure it is in top shape.
New or nearly new does not mean a thing.
I've had a battery crap out within a year of being "new."