I am very new to all of this too but had some fleece previously and put that in to spin dry with no ill effects. I put the fleece into 3 net laundry bags and washed (in a burco boiler to quite a hot hand hot) and then spun it without taking it out of the bags.
Having said that I read somewhere that Ryeland fleece doesn't felt easily so yours may be different. It might be worth trying a small batch though to see what happens.
As for the bags I got them off ebay last year in preparation for this years fleece (nothing like being prepared
) I can't remember how much they were but I bought a batch of 10 and I am sure I can't have paid too much for them. I just searched on polypropelene sacks
It seems that lots of others use old pillowcases too. I recently bought some (for rag rug making) at 50p each from a charity shop.