My coasters are stuffed with regular batting also, and I've never had a problem with water marks. However, if I had really beautiful furniture, and was worried about the water mark possibility, I might cut up a plastic placemat and slip pieces inside the coasters. There would still be the possibility of the condensation wicking through the fabric around the edges, but if you are paying attention to your drink, you would probably notice the coaster getting too wet and just swap it out with another coaster as needed.
I would probably drip dry when laundering.