Dear BJC,
First, I would say that once you have aimed the putter face and adopted your setup with reference to the putter face orientation, your only task is to make a stroke that sends the ball straight away from the putter face, with appropriate distance control.
Second, because of the first, this means that your eyes have a very limited role to play in making the stroke described above -- with your body square to the putter face's aim, you will always make exactly the same sort of movement with your shoulders, arms, and hands, and this does not really require ANY vision at all. Moving this way and how it feels to move this way is what you should focus your learning on, and not how the eyes might help or hurt in moving this way. The way I teach making the movement, you move your lead shoulder socket straight down at the balls of your lead foot, like a policeman would send a side-on battering ram back and up along a straight line along his feet. Knowing how to make the putter head move means you can do this move with the eyes closed.
Therefore, third, setup to a putter face and just make strokes with your eyes closed and focus on how your body feels to stroke straight thru the putter face. I would suggest setting up to five oir six foot putts without break (on a carpet or green), square the putter face, setup square to the putter face, and then just close your eyes and move the putter the way it needs to be moved. You can use a 9-foot putting mat for this purpose, just hitting the same putt over and over with the eyes closed or just concentrating on how it feels to always make the same movement in putting.
Then, fourth, setting up to a straight putt, think about how useless your eyes are once it's time to pull the trigger. So just sort of zone out of your vision even though your eyes stay open -- just like staring blankly into space (even as your eyes are directed towards the ball and putter). I like to look at the gap between the back of the ball and the putterface (maybe an eighth of an inch wide or more), or look at the grass between the ball and putter face at the bottom of the gap. Thjis sort of gaze tends to stabilize your eyes.
Next, fifth, just before you actually start the stroke, think of the feeling of your lead shoulder socket and see it in your mind's eye moving down at your foot to start the putter back. When you think of this feeling of the shoulder, your attention is oocupied with your body feeling, and vision just sort of idles in neutral.
Finally, sixth, an additional and complementary trick is to recognize that when your eyes return from a far target to the ball, in anticipation of starting the stroke back, the lens in the eyes has to refocus on the new closer distance to the ball at your feet. This refocusing is called "accommodation," and it really the relaxing of a ring of muscles that has been pulling your lens flat and thinner for far vision, so that (now relaxed) the lens puffs back to a rounder and fatter shape for near vision. This lens changing process takes a good one second to take place completely. So, during this time of gazing down at the ball before the ball comes into clear focus as the lens gets a little fatter, I just "wait for the shine" to appear on the cover of the ball. Once it does, that means my brain has sharply registered the exact distance from eyes to ball surface, and this focus clarity "stills" the eyes and helps keep the pivot of the stroke in the base of the neck still during the stroke. So, I look back to the ball and "wait for the shine," and once the shine is clear and sharp, and my eyes are nice and still, and my pivot nice and still, then and only then do I start the stroke back with the shoulder. Using this approach, I have no desire to move the eyes to watch the putter going back. I just continue looking straight down. The result is that the eyes keep gazing straight down while I am focused on the feeling of moving the shoulders.
The last tip is just the same as to place a dime beneath the ball and watch for the dime to appear right after impact. This also tends to keep the gaze still while the stroke takes place.
Let me know if this helps.
Cheers!
Geoff Mangum
Putting Theorist and Instructor
The PuttingZone.com
http://puttingzone.com
Golf's most advanced and comprehensive putting instruction.