I love teaching putting. Your issue is very common. One drill that I use is to put a tee in the ground so that the top is flush to the top of the turf. Place the ball on the tee and as you complete your stroke, mentally register the tee before you look up.
Another drill I use is to practice putting from 2-feet. Pic a straight putt so you don't have to worry about the line. Go thru your normal routine. Just before you start the stroke...Close your eyes. You will feel any flaws in your stroke.
My theory is this. We are naturally very curious. The first thing we learn in golf or any other sport is to keep your eyes on the ball. However, we wish to see where it is going before contact. Eliminate the feedback from your eyes, and you will train yourself that your stroke will send the ball on the line you have chosen. It is important to set the clubhead behind the ball so that it is square with your line. Then, address the clubhead. When you do that, the only thing left is speed.
I hope this helps.