You are correct! Had the replacement fone for less than two days and symptoms started happening again!!
I experimented with removing apps/widgets all with same end result. This morning I did a factory reset from recovery mode and did not restore a single app, so the phone is 100% oem but I don't have high hopes of that solving the issue either. I don't know what I am going to do. I wonder if they would let me switch to a different brand phone? A guy on XDA claims AT&T is aware of this and it is a programming error and only effects the S3 but no proof other than a conversation with someone at AT&T. He further claims he was told that rooting and a rom will also solve it but I am afraid to do that in case it failes and I have to go back to AT&T for help. I have heard stories of loosing the IMEI number during rooting also tripping the flash counter and may just need my warranty.
Forgot... You mention goikng down to 3g. Most of the time, I have all my radio's (GPS, WIFI, DATA) off and only use them on demand. I wish the answer was as simple as going to 3G.
Are you running Jelly Bean?