Are these first shear fleeces? They tend to hold together less well than from older sheep.
When you say a hand thick, do you mean across the palm so about 3-4"? That will be perfect for spinning.
If you are really worried about your fleece felting you can wash it slightly differently but it takes ages. Take a lock or two and hold it by each end (you will have to find the tips in your web then the butt end, rather than it being self-evident). Pull it tightly between your two hands and waft it around in the water - it won't felt because you are not letting the fibres become mixed up. Then waft it in the second bowl and that should be enough of a rinse. Lay it flat to dry. It's a bit tedious but it might be worth starting like that to see how you get on.
There's no time limit for washing although the sooner the better, but skirt/dag your fleeces before you put them away. For rolling, lay the whole thing out on a large board - 8'x6' then rearrange it by pulling the edges in and making it take up less space - I'm not good at describing this
It's a web because it's all stretched out sideways so for rolling you need it to be pretty much the same size and shape the sheep is.
However, being so easy to pull apart means you could probably spin it in the grease straight from the fleece, no washing or preparation at all. When spun in the grease, you wash it afterwards as yarn, when it doesn't felt.