I use a laundry basket to tote fleece around. This is helpful if I start it in the tub and want to move it to the sink area, because the basket will hold water that drips out of the fleece. The laundry baskets get hosed down outside and wiped down with bleach water before thy go back into regular service.
Note also the towel on the floor in front of the sink. I have yet to wash a fleece without some splashing.