Do you mean the large black connector that plugs into the rear of the left-side fuse panel (LBEC)?
My wife's 2005 Tahoe Z71 is experiencing similar issues. It will be fine, turn it off and then back on and the 'Service 4WD', ABS, BRAKE, indicators are all illuminated and the AC stops working. Upon further inspection the AC works (compressor engages) but the blower motor doesn't get power. The AC head unit is not getting 12vdc+ from the panel to switch the blower, instead I only see 1vdc+ when the truck is running and 0.085vdc+ when the ignition is in ACC. I moved back to the LBEC and found that all of the fuses in that panel were seeing the same sad switched voltage input, while the engine panel and battery were showing a steady 10-11vdc+. That makes me think that my issue (and the many similar ones I see reported on the various GMT800 forums are all the same, wiring or grounding issue with the LBEC. Did you see the same sort of conditions during your testing?
What is weird is the alternator works, start the truck and we get a nice 14.1vdc+ but the circuits don't come right. Without a full wiring diagram and systems operations manual for this damn thing I can only speculate but
During my diagnosis the battery died and had to be replaced, not shocked since it was five years old. After replacing the battery the truck came right -- sadly after a drive it went back to the same issues. At this point I'm certain it must be a wiring issue going into that panel, I looked under the dash and I don't see any worn wiring to cause shorts. I already checked the G110 and other related ground under the chassis -- they were caked with dirt but not corroded at all, cleaned and lightly sanded them just to be safe and re-torqued them to the frame, again to no change. I checked resistance to the engine block ground and alternator grounds to the battery, both read less than .5ohm which should be good.
Anyone have a picture/diagram of all of the grounding locations in these beasts? I'd happily ohm out each one just to clear grounding as an issue.