I would definitely use the old filter, or put all its media in the new filter. Since the new tank hasn't been cycled the fish really need it immediately. An uncycled tank is more likely to kill them than any diseases you might transfer over.
If you need to clean the old filter media, rinse it gently in a bucket of used tank water or dechlorinated water. Never wash filter media in water straight from the tap as the chlorine will kill the beneficial bacteria.
If you need to clean the old filter media, rinse it gently in a bucket of used tank water or dechlorinated water. Never wash filter media in water straight from the tap as the chlorine will kill the beneficial bacteria.