I changed my procedure slightly based on something else I've read. This time, I let the fleece presoak in a tub of room temperature water for an hour before actually starting to wash it. This gets off a great deal of dirt (but probably very little lanolin). The first batch of fleece that I presoaked seems to be much cleaner than the first batches I washed, but it also may be because I used a different cleanser. The first few batches were washed with "wool washes", stuff sold to wash wool. These are expensive. But then I read that you can use almost any sort of cleanser... including dish detergent and shampoo. (Clothes detergent was discouraged though, because of all the added brighteners, etc.) So for the second batch, I'm using Suave Shampoo at $1.99 a bottle (vs. Kookaburra wool wash at $10 a bottle.)
It'll be several hours before this last bit of fleece is cleaned but I will be very glad once it's all done!
Now to finish staining my spinning wheel so that I can assemble the darn thing.