Blurred Lines on screen, then computer shuts off
I went to Apple Care and they quoted me $700 to fix the issue, which I think they said was the logic board...Is it likely that the logic board is the real problem?


If apple ran in house diagnostics on the machine and told you that you had a faulty logic board, then the logic board needs to be replaced. The issue could specifically be the with the integrated graphics component of your CPU, but it's a moot point since it still requires that the logic board needs replacement. 

This is a hardware issue for which no troubleshooting can ameliorate the problem or make it disappear. This problem will only get worse; inevitably (and without warning) the logic board will completely die and the computer will not boot. 


  Is it worth investigating/replacing myself?


That's subjective. According to iFixit, the OEM part costs either $400 or $600 dollars depending on which processor you have. In addition, factor in the cost of necessary tools and difficulty of fixing it. Draw your decision based on the cost/benefit of doing it yourself vs. taking it to Apple. 

Best of luck.