of course 6970 is the 2nd fastest gpu in ati brand.5970 is a dual gpu but it can't beat 6970 because of it's higher clock,stream processing and texture unit etc by the way 6970 is recommended for bf3 and oh,eyefinity is fine when you get an 6970.
Maybe it's reccomended for BF3 but in general the 5970 is faster and you stated the 6970 is allways faster. I googled for 2 minutes and saw benchmarks and comments and stuff all saying the 5970 is the fastest.
It's possible that the VRAM is causing some issue but most likely it's the same issues a lot of people are having. Some days my CF 5850s run the game at Ultra just fine, other days it stutters and has texture problems. I have a 2nd monitor which I display a lot of info with a custom Rainmeter skin I made, so I can watch my FPS, GPU usage and temps, as well as CPU, RAM, etc. Anyway the point is that quite often the 2nd card's usage just randomly drops, and that's when the game starts to get all buggy. I'd be willing to bet it's the same for you, OP.
The only reason 6970 is recommended is because it is from their new line up. Why would they want people to buy there old line up? Yes, the 5970 is faster, but people tend to buy the 6970 because it is new and it probably does not have that much heat issues or whatever.
Just because its new, doesn't automatically mean its better.