You need to edit your APNs
1) Going into Mobile Networks and manually inputting your carrier's APNs
2) adding apn's
via this method
3) Maybe try step A3 from here
. If you decide to do steps B1-4, ignore any mention of Phone Info and use Settings/Mobile Networks/Mobile Networks Settings/Preferred Network Type to toggle Global (as you've probably already done)
I have absolutely no experience with any of this. I've read a good bit on this as I've been on this phone.
[VZW] Samsung Galaxy SIII