Random BSODs for a few months

The oldest driver I can find that's compatible with GTX 770 is 320.18

Hmm, You'll have to try a few driver versions then if you can't find a stable one.

I would start with 320.18

I got about 6 BSODs after installing 320.18 but I only got dumps for 3 of them.

Does changing the driver correlate with the amount of BSODs you get?

Can you try running MSE full scan as well.

Stress test your CPU with Prime95

I recommend running Prime95 for around 2 hours

   Warning

As with most stress tests your component's temperature will rapidly increase.
It is advised that you download a temperature monitoring program like HWmonitor
Keep an eye on your temperatures and stop the tests if it starts to overheat

Well usually when I update the drivers the BSODs don't increase but this time they did. My computer has gotten extremely slow that I needed to do a system restore.