The nice thing about strings is that you can use any size (length and width). I don't use a stabilizer.
I also make my own "fabric" by simply sewing together the odd pieces of fabric. Then I straighten one edge and add to it. The piece just keeps growing. When "done" I use it to sub-cut into pattern pieces.
As for throwing away scraps - shudder my heart. I only toss true scraps - something smaller than 1"
Recently I saw someone post that they sub-cut all their left-overs into strips and sorted the strips by width. They cut @ 1 1/2, 2, 2 1/2 and so on depending on the widest cut they could get out of a piece. Then when they were looking for something that width, they would consult the bin first.
Check out Bonnie Hunter's scrap taming method. She works primarily with scraps.