It's not really important but if the device was branded as AT&T it isn't an unlocked device. When we discuss unlocked devices typically it means that there is no carrier branding, preinstalled carrier apps or preinstalled carrier firmware. Most of the big carriers release/unlock devices after you've met their service agreement, so in essence the phone isn't truly locked, like in the days of old, but then becomes a carrier unlocked device. It still has all of the branding, carrier apps and carrier firmware but if you want they don't block you from activating the device with another carrier.
It sounds like you have a carrier unlocked device, since you needed to get an unlock code from the carrier for it to be released. It seems likely this is the issue, AT&T firmware with a Verizon SIM card. Some have mentioned that simply installing the new Carrier's SIM card changes the device, which I think might be true of AT&T but with Verizon you'll likely have to have their firmware flashed to the device. Verizon Corporate stores can do this and I'm pretty sure they do it for free. Authorized dealers may be able to do this also but I've heard some of them charge.
Where I'm a little confused is, where could you go that Verizon doesn't cover? I mean if the device works in your hometown, I would expect it to work the same several states over since they are the largest network.
If the device is/was branded AT&T I'd take it to Verizon and get face to face help. Their over the phone help is sometimes lacking.