Yeah, I think swapping cards will help narrow down the possibilities. Sounds like you've really gotten the short stick here -- but statistically someone was bound to get screwy errors at some point on this forum, things have been going smoothly for too long

Just stay positive about it.
A format may or may not help -- the sucky part is that you'll have to re-download all those games again. It might be a good idea to re-format and install Windows fresh (or get an old HDD that's laying around and use that, then put your current one off to the side just in case it's not a Windows error -- that way you don't have to re-load
everything). Download / install one game for now, whichever one most reliably crashes it. If it still crashes, try your friend's card and that will tell us if it's a GPU issue.
I'd hate to have you re-format and find out it's a hardware issue, though. I guess it has to be done at some point to ensure it's not something worse (like a motherboard issue).
Are you getting any video artifacting when you play games?
Re: beta drivers, they're just a more recent iteration of video drivers. It looks like the 6870 does not currently have beta drivers available.
support.amd.com/us/gpudownload/windows/Pages/auto_detect.aspx
Try that tool to make sure you've got the right drivers on there (this is a stretch, but it'd be good to neutralize one more element). If you've already ensured they're correct, ignore it.