If your coach isn't level after using the AUTO LEVEL system, that can be adjusted. I'm not sure what jacks Anthem uses, I'm guessing they're HWH. Page 21 of the HWH Manual (below) tells you how to adjust the level sensor.
HWH 725 MANUAL.pdf
To answer your initial question, yes, you can start the leveling process over again when the slides are out, and the shore power is connected. Actually, you should be connected to shore power before leveling.
You say you don't always here the airbags dump when leveling. If you're doing a re-level, they've already dumped, and you won't hear them again. However, on the HWH system, when you AUTO LEVEL, the system has a timer when it dumps the airbags, since it wouldn't actually know if they're dumped. If you AUTO LEVEL a second time without airing up, that timer still runs and there should be a lull before the jacks start moving, waiting for that timer to time out.
You should try and use AUTO LEVEL when you can. At the end of the auto leveling, the system checks that all jacks have pressure on them (touching the ground). When you manually level, that doesn't occur.
Lastly.....again, I'm not familiar with the Cornerstone level, but doesn't it have air and jack leveling. Are mixing up the two?