Ok, check this out, I've repaired around five of them on location:
If you look at the center of the BNC connection on the camera, you'll see the positive post; have a closer look and compare it to the BNC just above it. Is there a difference with that little "4-piece tube" protruding from the center of the connection? Probably YES. Those 4 parts of the circular "tube" are split abnormally compared to the one (never use, I think it's a Genlock BNC; don't remember and don't have a cam in front of me).
Now take a small screwdriver and just bring pack those 4 parts of the circular "tube" towards center.
You fixed it, no more E/F alterning on the screen, solid "refurbished" BNC, thanks to the soundguy on the spot.
The first one I caught, one of the 4 parts of that positive connector was completely crushed at the bottom of the BNC and was creating a short from center post (positive) to the outer shell (negative) of the connector ! Since then, I simply always check those connectors on the cam since those C300s are getting older and older now.
As a side funny exercise, check out the Sony's FS7s connectors now. Wow, completely different model. Way more solid.