Given that this seems to have started happening after a hardware change and occurs in two different (though related) operating systems, I'd be tempted to go back to the previous hardware configuration (put the old card back in place of the new one if you still have it) and see if the problems remain or go away. Alternatively, borrow another card from someone else, or try your new card in a different machine. Is the new card still in a period where you could return it and get a replacement?
How good is your power supply unit, and does it provide enough power to run the system with the new card? That's just popped into my head, that perhaps the new system draws more power at times than the PSU can supply, but I don't know what the visible effects of that would be. I guess the voltages would be off when that happens - on my system I can display that on a monitor within Windows.
If you think the drivers are to blame (which could be possible if you're using the same drivers on Win7 and Win8), could you try uninstalling them then reinstalling an older version?
While it usually happens when playing a game (any game), sometimes the computer will just start up and immediately show the signs that it will crash soon
Does that include cold starts where the computer has been off for a while beforehand (so that everything has had the chance to cool down)? If so, for me that would rule out a temperature issue, as the card shouldn't have time to overheat...