Whether to dry the cardboard or not depends on the situation with your compost pile. If the compost pile is already dry, then there's no need to dry the cardboard. You can simply add the soaked cardboard directly to the compost pile. On the other hand, if your compost pile is very moist, then you might want to dry the cardboard first. The reason for this is that wet cardboard will absorb more water than dry cardboard. Soaking the cardboard will remove some of its moisture content, which will make it easier to shred.