Cardboard is bad for long term storage but I wouldn't be too worried myself for short-term sorting.
I use a plastic sorting box very similar to yours, a little larger than this Craft Organizer from
Amazon.

I have it standing upright on my desk against a side wall, like a miniature bookshelf, for sorting and temporary storage of recent purchases in different categories.
I used to have two 12-drawer storage organizers from the hardware store, meant for holding different sized nuts&bolts and screws etc, and that was fine except they took up more space, and opening and closing the drawers was getting tedious. I found mine at The Container Store but can't find the same one online anymore, as I'd like to buy another.