You're most likely getting grainy footage because Twitch's Bitrate limit for non-Partners is 6000kbps which makes for pretty low-quality 1080p60fps footage. Bumping your resolution down to 720p (or 900p if you really want to), would most likely make things look better. Keep in mind most people watch on their mobile devices and 720p60fps actually tends to look better at the 6000kbps Twitch limit.
But you didn't post what encoder you're using (x264 or NVENC) or Bitrate so I can't definitively say. You also list a capture card, are you feeding your gaming PC into the HD60S or are you using it to capture console?
From the PC specs, it really depends on what you want to do. The GTX 1080s could handle using NVENC either using 1PC or 2PC setup. The i7 8700k is the more powerful CPU out of the two, so it just depends on if you want better gaming performance or better video quality. It would help to know what games you're streaming to give you a better suggestion.
If it were me, I would probably use the i7 8700k/GTX 1080 as my Streaming PC and the AMD FX-8350/GTX 1080 as my Gaming PC, but making sure that the Streaming PC is ONLY doing the encoding. That means in OBS on the Gaming PC have the webcam and any overlays or graphics on that PC, and then feed that into the capture card on the Streaming PC. On the streaming PC use x264 encoding with the 'CPU Usage Preset' set to as high as it will go without dropping/lagging frames (probably 'faster' or 'fast' preset) and set the 'Output (scaled) Resolution' to 720p or 900p. Then set the rest of the settings (ie. your bitrate, keyframe interval) according to
Twitch's Requirements.
That should get yourself in a better place. But make sure to update us on what games you stream, what bitrate you were using, and if you are indeed using the capture card for capturing the gaming PC! Once you do that, then people can give you some more personalized suggestions!
