Instead of buying pre-cuts, you might want to consider buying batting by the yard to avoid so many odd off-cut sizes.
When I have larger pieces, I measure and fold back up, with a label as to the size. If it's an odd shape (eg. L-shaped) I leave it as is, and include a sketch, with the measurements of all the sides. Then when I shop my batting stash, I know what is there waiting. Just be sure to include what kind it is along with the quilting distance.
For storage, I have one of those huge trunk-like Rubbermaid totes ... but it always seems to be overflowing. So am now considering other options, though it may just be a 2nd tote!