Thanks for that extra detail, elcodora. I have actually seen the message that reads "VZWAVSServices are not responding or can't be found".
Do you happen to have Google installed? If so, please go to the app to either uninstall the update or turn the app off entirely until the Google app is updated by Google to work with the newest software version on the SIII. This is just a temporary work around until this can be fixed.
As far as the internet connection with WiFi, please go to Settings>WiFi>Menue Key>Advanced and uncheck Auto Network Switch. This is the feature that causes a negative effect with the WiFi networks you are connecting to.
Please let me know how the device works once these steps are completed.
That did not work to access the data. It still says I can't connect. My wife got a new phone for Xmas and used my upgrade. Could some thing have happened during that switch? I keep trying to think what could be causing it but can't figure it out since everyone else in my household can get data.
Have you tried reseating the SIM card? In other words, turn off the phone, remove the SIM card (be careful not to touch the gold contacts), wait a few minutes, then reinsert the SIM card, turn the phone back on, and allow it a few minutes to establish a fresh network connection. Also ensure that your mobile network is set to LTE/CDMA and not Global.