Try updating the TV firmware. I think most games run at 60hz. Something is really messing up. Try another lead and maybe try the console on another TV. I have had the hdmi fail before on ps3 but that was a terminal issue. It wasn't a cable.
I'm sorry I can't help much more
Oh I have actually came across an hdmi cable that supports ethernet through it. It has glowing plugs
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How do you update a tv? My tv is not a smart tv.
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I'm all out if ideas. Google your model and see if anyone else had issues. Try another TV as it's odd it's both consoles. A new cable means nothing. Make sure it's a high speed hdmi
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On my tv has three HDMI plugins and my bottom two was the problem of the shaking thing. I had my xbox one on the bottom one.