My 1-2 yard cuts of fabric I fold into 1 yard lengths. Then I fold in half to 18", then into thirds. I fold the top down 2/3 way, fold the other half up and secure that inside the 2/3 folded edge. Stack so neat on my shelves.
For larger cuts, 3 yards and over, I prefer "fanning" the folds. 6-7" is face up, the next 6-7" folded the other way, Fold next 6-7" the other way, etc. That way, I can lay the fabric with the long fold to my left and take off the top only what I need. They still stack the same, but each fabric is just a 'taller' stack of fabric.
I have never used the cardboard. When I fold every piece of fabric like this, I have neat folds of fabric every time, all very close in width. The fabric sits in stacks in closed cabinets-5 stacks to a shelf. I don't fret (anymore) if a fabric folded is an inch or so larger than the others on the shelf. I have learned to let that go.