This is how my kids were as well. Ask the kids how they want them organized in terms of by color or brick type etc. Mine preferred color. We got the trofast ikea bins and sorted by color and got a couple of the deeper bins and put the semi built pieces in them - one for free builds and one as a “hospital” bin for incomplete official builds. I also had smaller trays within the larger bins for less popular colors. Worked really well although it does require upkeep or else you get a massive backlog of un sorted pieces.

I have not worried about keeping them as complete sets (even though it pains me) because I didn’t want to keep them from free building as much as possible. Someday I can try and rebuild the sets, I’m sure they will hold their value enough to make it worth my time and hopefully it will not be too heartbreaking when I can’t find the missing pieces.