iPhone 6S visual voice mail only works intermittently. Help please!

11.29.16

I have an iPhone 6S running iOS 10.1.0 on AT&T.  Both  Apple and AT&T software are up to date. When I try to listen to visual voice mail by going to the icon in the phone app, I can see there are voice mail messages but when I press to hear the message it goes to the screen like it is going to play the message it may or may not play 1-5 seconds of the message then it stops and pops back to the previous screen showing me all the voice mail messages. 

It happens intermittently both on Wi-Fi and LTE.  Apple says it's an AT&T issue. AT&T says it's an Apple issue. 

So far I have both hard and soft reset the phone. Restored the phone to factory settings and set up the phone as a new device. I've tried switching Siri to various accents and restarting the phone. (Someone wrote that idea)  

I finally gave up and actually ordered a NEW iPhone. It arrived, I set it up as a new device and had AT&T switch phones on my account. I installed the backup I had saved in iCloud. And the new phone has the same issue. 

AT&T store tells me I have to go through calling tech support AGAIN and get to a level 2 tech who will then check the AT&T end on my account and "reset the sockets" it could take 10 days. When I call tech support they don't want to hear what I've already done to try and fix it. They want me to reset the device or turn off Wi-Fi etc. They don't listen to the fact that I have a new phone that is doing the same thing as the one I replaced. They don't seem to listen to anything I try to say. It seems like they have a script they use and won't deviate from it. It's ridiculous. 

I've have had to reset the phones so many times, I'm about ready to leave Apple, AT&T or both. The hours I've sent trying to fix this is crazy.

Not installing my backed up info is not an option. 

Anyone know how to fix this or get directly to level 2 or 3 tech support?

Thanks,

~ At My Wits End