So uhh, graduating media student here, with adobe ch problems.

I am not sure, but some suggestions. 

I would try adjusting the light on your face. In the video one side is a bit darker than the other - I was wondering if that is confusing it (in which case you will find nothing i. the puppet itself!)

Blink layers normally fire instead of using eyelids to close eyes. I wonder if actually you don't want blink and just use eyelid closing instead, so you have finer control. But that would require different artwork.

I was not sure why you had the cycle layers pause tag on one of the layers in a screenshot. I don't think you need that.

You could but a keyboard trigger on the blink layers and try turning off the camera. Does it behave correctly then? If not, then it's not the webcam. Might help narrow down the problem.

I personally would go through every layer in the rigging hierarchy and check all the tags. A stray tag or a Left tag on the right side can causes havoc.

There is nothing wrong with that page you mentioned, but here is a blog i wrote a few years ago in case anything useful in it. https://extra-ordinary.tv/2018/04/21/debugging-character-animator-eyess/

But the fact that things work better for one eye than the other makes me think lighting or stray/wrong tags.