I'm in the other boat - it seems like where I am (central Indiana) there are actually a few Verizon models readily available and AT&T ones are in short supply. The one Walmart that I sometimes visit had both a white 32GB and a black 16GB in stock the other day, but they can't keep AT&T ones at all.
As for the Apple Store, I haven't really looked here for Verizon ones, and I know Target wasn't allowed to do preorders with Verizon ones (the lady there overshared while I was looking at their case selections), and Radio Shack seems to be getting them out to people as soon as they get them in (still have a few preorders of their own, but mostly AT&T).
Not sure if this helps, but I think it varies by region and carrier a lot.