Generally filters used to be available at the resort 'gift' shops in the lobby, rather than the general store. If you tried the one at Boardwalk, maybe check over at Beach or Yacht to see if theirs has any. Could be they're just not stocking them anymore.
Paper towels can work in a pinch - single layer can be pretty flimsy and rip easily plus lets the water pour through too quickly - but be careful with double layer paper towels as the flow can be very slow, so much so that the basket can overflow with water - best to use smaller amounts of water at a time - a few cups at a time, rather than all at once.