I'm another that turns on the full EB at startup, particularly since moving to GA where it's a lot hillier than AZ. I don't use tow/haul though when unloaded. Although the EB will work better in T/H, it doesn't drive as smooth unloaded.
As for operation, full is engaged anytime you're off the throttle and the T/C is locked (even when not on the brakes), and disengages at low speeds (others say low engine RPM, which may be the case). Auto is also engaged when you're off throttle and T/C is locked, but it will attempt to maintain the speed you were at when you got off the throttle. As was said in the case of cruise control it will attempt to maintain that speed.
The operation described above is new for the past 3 model years I believe. The prior model years with E/B had a slightly different function, with multiple settings. Figured I'd point that out in case others with older trucks describe how theirs works and it sounds different than yours.