I have always used shavings....and touch wood, never had a problem.
With that said i am switching to fleece, which I'm hoping will obviously reduce cleaning times as it takes me a good couple of hours to get piggies out, empty, sweep up, clean and wash down, dry etc On colder days i use my dads hot air gun to dry the hutches out.
In summer its fairly easy to put pigs in run and get on with it, but with the weather turning a bit temperamental fleece will help hugely!
With regards to the wash bags, if a pillow case isnt big enough, what about a childs single duvet cover? Put a zip on end and voila....

There is a lady locally to me who makes and sells a wide variety of fleece items, ranging from waterproof lap pads, cage liners, cozy hideouts etc aswel as the wash bags, bottles,and more. She has a facebook page called snoozie's cozies, and all mine love their cozies....and she ships worldwide as far as i know