In putting, the fewer moving parts there are the simpler and more consistent your method will probably be. To that end, I encourage you to work on building a stroke that is controlled by the shoulders.
To promote this pendulum-type action, try the following exercise. Take your address position, and place your palms together in a downward praying position. Focus on the triangle that is formed between your hands, arms and shoulders, and then, keeping your head and lower body perfectly still, rock your shoulders so that you move the triangle smoothly back and through (1). Create an under-and-up pendulum motion with your shoulders, where your chest stays square to the target line, as opposed to opening your body to the hole. Feel that your left shoulder works down on the backswing, then up on the throughswing. This will control the path of the putter-head and keep the putter on line through impact (2).
On the follow-through - particularly on the longer putts - allow your head to rotate down the line towards the target after you have completed your stroke. That enables your triangle to work correctly, and thus allows you to follow the progress of the ball all the way to the hole without lifting your body out of position.