I would not TRY to keep the handle of the putter anywhere. Remember it is easier for the hands to move to a place rather than stay in a place. The mere thought of keeping the handle steady is throw-away. Your words convey the "steering" that you are going through. And the only thing you are suffering from is a bent plane line.
Do two things. Find a bench and move the shaft of your putter along the bench. Don't worry about where the handle is. Just move the putter back and through. Next hit putts with with two pennies two inches behind the ball. Don't hit the pennies. This will stop the throw-away of the club.
Mike Adams gave my a practice drill similar to the VJ mentions.
Stack 3 dimes a putter head length behind the ball. Pick off the top dime on the backstroke and do not hit either of the 2 dimes on the forward motion.
Hybrid Mode
