Hey Luke,
As described, your problem seems to be your RF connector. If your console was "blinking" with the TV I would say its a pin connector issue.
Your fix should be relatively easy.
[image|937115]
On the top of the image (right side of the console) you should see a video and audio jacks (yellow and red) which correspond to the [https://en.wikipedia.org/wiki/RCA_connector|RCA connectors] (or composite) jacks on your TV. There is no white because the NES was mono rather than stereo for sound.
These connectors are very common and rather inexpensive. Try them before you dig into the [guide|3822|72-pin connector refurbishment].
Hope this helps.