I think for you to achieve proper putting alignment when playing you need to work on your alignment when practice putting.
There are some great training aids for putting that can help you with alignment. The Eye Line putting mirror, the Dirty Larry putting aid attachment, string tied to 2 Tees or simply alignment rods.
A simple way to practice is to set your alignment rod up in the direction that you need, set your ball down with the alignment line running parallel to your alignment rod. Practice getting your body parallel to the alignment rod. Roll some putts trying to keep the line on ball rolling as tightly as possible, you don’t want see the line wobbling.
Now when you get out on the course you can use the line on your ball to help you aim, but it can also help you to align your body. From here try and make the ball roll end over end in a tight line just like you did in practice.