I have several sets of the Piggy Bedspreads and the instructions say to never use any type of dryer sheets with them. The sheets will leave a coating on the fleece that will make it lose the wicking action.
I have the laundry balls from Grove Collaborative that I use for the clothes in my dryer but I've never used them with my fleece. I looked at the product on Amazon and the only thing that would make me not want to use them is the instructions that say don't use them for fluffy clothes. I don't know if fleece would be consider a fluffy product. I know how frustrating it is to get all of the hair off of the fleece liners. I take my fleece outside and shake it into the yard but I believe you live in an apartment and I know that isn't always an available option.
To be honest, I've sort of given up on removing all of the hair from the fleece. I use a brush and then shake it the best I can and then I just hope for the best. I have the horse blanket bags and they do work pretty well at containing the loose hair but I don't always use them. If you have a large enough washer that might be a good option for you.
As Guinea Pig Papa said, fabric softener sheets or liquid will cause issues with absorbency. If hair is really imbedded in the fleece, you may want to try a different type of fleece, such as the anti-pill plush at Joann Fabrics. It has a smoother texture, so hair and hay don't stick to it as much. I have some of the cheaper blizzard fleece and that pills and everything seems to stick to it.