If it was me I would first move as many of the cart type organizers out of the room just as they are for now. Then I would make a place on the floor to pile everything from those white wood shelf cubbies Once they are empty assign each one a color and start folding/sorting to them. Use the cubbies as your control center. If one fills up then move the full contents of that cubbie to another area even if you need to move it to a spare bed. Once you have that pile on the floor under control move to the wire cubbies and empty them the same way but now you have an area to place the folded fabric back into the room.
You might want to get a large box(es) to toss notions and goodies in till the fabric is tamed. Then you can empty and sort the notions. I would empty all of the cart things together then sort once.
Good luck and happy hunting