You're not doing anything wrong by starting the cabbage seed off in 3" pots. I do the same. The problem you are experiencing is caused by leaving them too long. By the time the seedlings have begun developing the first true leaves they need much more space, particularly depth and should be transplanted into either bigger pots or deep boxes.
After about two weeks or so they are then moved again either to the great outdoors or a large container as suits your situation.