I uninstalled everything again and used driver sweeper this time around. I reinstalled the AMD drivers again and ran Heaven. The drivers no longer crash and the stuttering is much better than it was before!
Since there was still some unusual stuttering of the graphics and sound I tried underclocking. It did not effect the stuttering at all.
I didn't uninstall MSI Afterburner because it works on most brands of cards, not just msi cards. I also tried MSI's Kombustor for stress testing and there was no jitering, stuttering, or hanging. I tried both DX11 and OpenGL and stutter on Heaven but not on Kombustor.
I should also probably mention that I have Heaven installed on an SSD so loading shouldn't be the cause of the stuttering.