I can't get the hang of washing whole fleeces at a time, I wash my fleeces in chunks that'll fit in a washing-up bowl. I use the net laundry bags, fleece in that, in a slatted basket that fits inside the washing-up bowl. So I can lift the fleece in and out of the washing-up bowl without disturbing it unduly, and drain it ditto.
Having had a scare where I thought I'd clogged the washing machine pipes with lanolin, I recently bought a salad spinner, which conveniently takes exactly the same amount of fleece as above, so I can spin-dry it by hand

. As well as getting it part dry after last rinse, I can also spin out the water at each stage of washing, which reduces the amount of rinsing I need to do.
The pic shows using the net bags and washing-up bowls to give the RBST Norfolk Horn a cold soak - and you can see how clean just an overnight soak gets the fibre. I span this just like that, washing the last of the grease out after plying.