Can you show the rigging hierarchy of the eyes? I suspect something is not independent when it needs to be.
In case helpful: https://extra-ordinary.tv/2018/04/21/debugging-character-animator-eyess/ - the most common problems I have seen (and how to fix them).
Here it is. I also see there are Rigging Issues with the Left and Right Pupil. I've posted those below. I suspect its because my eyeball as I've drawn it is much smaller than the typical Pupil Range. But I don't know how to limit it.
I added a Pupil Range and its a little better. Now when I look ahead its normal but when I look side to side, the eyes get distorted.
Those warning messages makes it look like you added handles somewhere (on eye layers?). Delete those handles. If still not working, have a look at the eye gaze behavior in rigging mode and expand the handles and views etc sections. They are the layers the behavior is really using. Plz share that screen shot. Maybe it's picked the wrong layers due to stray tags.