When I traded in my iPhone for an S5 last year, I couldn't get my voicemail set up for a week....not even the vm that you call in from the dialer. I forget the exact message that Visual VM gave me, but it very well could have been the same as yours. Anyway, I took the phone back into my local AT&T store and when I explained the problem, the girl said they never activated voicemail for my new phone. A few quick clicks and I was good to go. I know you said you already called AT&T, but if you spoke to tech support, it's possible that they don't have the same view of your account like a salesperson would. Maybe AT&T forgot to transfer your vm over from the old phone to the new one?
Other than that, I can even begin to think of a solution to your problem. Only other thing that comes to mind is that maybe you simply got a bad phone. It sucks, but it's bound to happen with anything that is mass produced. Since you've only had it a week, I can't imagine why you couldn't take it back to AT&T and exchange it for a new one.