Not to worry most systems don't have warranty coverage, unless Apple has acknowledged they had a problem that they will service for free. I don't know of any extended warranties on this system related to the video.
In this case yes, replacing the logic board out is a viable solution here. Since you've already taken a dip into the deep end you shouldn't have any problems. Just make sure to follow proper ESD practices in the process.
If you got some extra cash see if you can boost your CPU speed on the replacement logic board. Just make sure it's compatible with your given system (maybe going up to an Intel i7).