Move to a NVIDIA card, you're still going to get driver crashes, just everywhere and anytime in the system instead of in a game. Why not pull one of the cards and see if CrossFire itself is the problem for the time being.
Also, a few other things to note:
You're overclocking. Your crashes may be due to the CPU.
Are you overclocking the video cards as well? If so, you know what comes next.
How is your case's air flow? You've got two fairly hot cards running in there, along with an overclocked CPU, a high capacity PSU, all generating heat.
Also, when you moved to your 4780's, what video cards did you have in there previously? Did you go from NVIDIA to ATI without cleaning your system properly from the old drivers?
In that vein, what OS are you using? If it's XP, I'd suggest doing a full reinstall of the OS if you've changed hardware.
There's so many variables, I'd say if you don't want to deal with a multi-gpu solution just pull one of the cards to start with and see where it goes from there.