in the book i read which is in part by a former touring pro it states that most players have at least some amount of cross swing in there follow thru. In addition i have kept practicing and realized that if i align myself with the object ball(hope that makes sense) more and "fall" onto the cue ball i have alot more success with my cue being perfectly straight....not every time tho.....before i was kind of using both balls and aligning myself with the cue ball. Not sure if that makes sense or if its right or wrong but it works for me and everyones visual and mental perception is different.
From what you said, I would take a wild guess & say that you might be cross dominant eyed for pool.
I have always done what you said in your second explanation while playing very well.
Then when coming back from a good while of very casual play & playing more seriously some misses of shots that I should not miss just started popping up from out of no where & I noticed that they where more when cutting to one side more than the other & I also noticed that I was putting a hair of right english on the ball when I did not want any.
I posted about it here on AZB & Gene Albrecht of Perfect Aim contacted me. He convinced me that I was left eye dominant for pool. He may know more about eye issues as they relate to pool than anyone else at this time & also why they tend to be so.
Anyway one of the manual methods is sort of what you say in your first explanation, looking at the OB the whole time until set & then & only then look at the CB... & trust what you have even if it does not look quite right. It almost always is & a miss may be due to steering the cue because you do not truly trust it & hence do not stroke straight.
Anyway give that method a consistent try or contacting Gene would be my next advice.
Good Luck & Best Wishes for You & Yours,
Rick
PS How is it said? The eyes lead & body follows. Well if the eyes see a crooked line as straight then the body will be aligned crooked.