It might be helpful if you could add a pic or 2 so we can see what the situation is. If there's grass growing, poking holes in your smother will defeat the purpose. Hopefully, under the shadow of shrubs and trees, there's not much, if anything, that needs to be smothered. If so, pulling those plants out by hand, and then placing the smother around the plants, but leaving a gap around the trunks about the same as the diameter of each plant should not prevent so much moisture from soaking in that it causes a problem.
With a thick layer of mulch/compost on top, the cardboard will stay moist after it rains a few times and shouldn't be a problem unless it doesn't rain for so long that you would have needed to water w/o the smother anyway.
Single-layer or colored or coated cardboard is not something I would use in a garden. I take all of that to the recycle place. Brown, corrugated cardboard is what I've always used for smothering, remove any tape or staples, and sometimes newspaper if there's nothing strong growing and it's a very small spot. Newspaper does "melt" a lot more quickly than cardboard. A mix of materials is a great idea if the growth situation is varied throughout the area.
