Hmmm... So you're AA never becomes active and is always listed as offline? Try this, sitting on the side of the rode with the car on and not plugged into AA. Do you have signal? Can you cruise the web?
Connect the USB and start AA on your car. Realize that your phone should connect to your car for the phone Bluetooth profile but for the media it should be going through the USB. Can you cruise the web now?
AA tries hard to not let you use other functionality on the phone while connected but there are ways around this. I've been able to hit the "Show Recent Apps" button, pick another app, let's say Calendar, then hit the "Back" button until you get to your home screen on the phone. Check out the data connection in settings. Are you connected. Start Google Chrome or GMail. Can you connect to the internet and browse to places or read your mail?
If you can then it's very suspicious why AA on the head unit of your car says your offline since using your phone directly shows us otherwise.
If you can't then what's stopping you? For example, are you connected via mobile data? What if you're in your driveway and are close enough to connect to your wifi? If you're connected to your wifi and cannot surf the web while also connected to AA that'd be weird.
Assuming you were able to do such internet things before you hooked up AA, what happens if you pull the USB cord and try to surf the web again?
Not sure what you problem may be but these are some steps you can do to perhaps gain some information about what's going on.
Good luck and report back...