I do, and I love it. Both mum and I are allergic to most bedding types, though we have found that having a small litter tray does come in handy under the hay rack.
Fleece is a very popular bedding choice, though it is high maintenance, you have to clean it several times a day and if you use the same washing machine for the fleece as you do for your clothes expect to find stray pieces of hay on them unless you shake it vigorously before washing.
Fleece would be pretty cheap now too, with it being Winter there would also be lots of different patterns and designs in fabric stores.
Newspaper as a bedding isn't a good idea, as you have found out, you want to keep the top layer as dry as possible, and with newspaper pee just makes it go mushy. It normally takes a bit of experimenting to get something that works for you, but currently I use a layer of fleece then a layer of towels.