Looks to me like you are going to be sleeping at DGMs house for a while if you want to get it done by August. All the advice above is good. I like the idea of getting lots of wash baskets and just start sorting and get the cottons in order color at the start. As long as the cotton get sorted into color baskets it will be easier to go back to each and start the folding. DGM can do some of the folding if it is ready for her.
Worry about all the other fabric later towards the end of organization.
Use one basket for all the notions as you find them and just store them in that one place.
Use another basket for thread.
Sometimes my room looks a lot like this and I don't think it will be as bad as it looks. The folding may take the longest time.
Please keep posting as you progress. I love to do this kind of work and if I were neighbor I would love to help.
peace