You're below min req. on that PC. Go with the new build, no point upgrading. FX 6300 is a very good CPU for this task. I believe it should beat by the Intel i3 Skylake, but I'm not entirely sure. I do know that the i5 from Skylake can match the FX 8300 series in rendering tasks though.
I'm confused why you want to use Premiere and Sony Vegas, they are both NLE's.
I recommend HitFilm 3 Express if you want the most from your graphics card, as its not software rendering on the CPU, instead its done using OpenGL which is faster. The program is 100% free. It wont matter whether you choose NVIDIA or AMD here.
I do recommend going with Intel though. The FX stock coolers are so loud, that's nitpicking, but still worth mentioning.
All the best!