Here is how I fixed my issue with my Radeon 6950, as it seemed to be a driver issue.

1. Completely uninstall video driver (13.8 Beta)
2. Reboot PC
3. Download 13.6 Driver, NOT 13.8 Beta.
4. Install driver
5. Reboot
6. Play Game

Before this, I tried everything from re installing FFXIV, changing graphics settings, and re installing driver. As soon as I went down one version and didn't use the beta driver, it worked perfectly and I haven't crashed since EA. For those with other brands and models of video cards, I would suggest the same thing. Downgrade the version of your driver or upgrade depending on which your on.