You can't go wrong with the S3. Pricing is very good right now. If you can wait for the S4 it will be probably be the best phone of 2013. Either way you can't go wrong.
Reboot phone
Turn off Wifi
Launch Verizon Mail.
Turn sync to auto.
Back out
Relaunch.
Hopefully that will work. My phone was acting up once I got the JB update. Everything seems normal now. Good luck!