Cardboard, for sure. Wood, not so sure about, it leaches "tannins". It is mildly acidic and occurs in bogs where famously you don't get decomposition, but that's at far higher concentration. Don't know the effect on worms or composting, I just know it from using wood inside an aquarium.

Cardboard water will give you some of the starch based glue they use to make it (that slimy stuff you can feel on the inside) which is a nice addition.

Personally if you soak the cardboard and wood in different containers, I'd use the cardboard water and not use the wood. Don't know any adv from the wood water, and potential risk or maybe slowing down decomposition of your bin. I use left over coffee liquid to wet my bedding/feedings down (worms love it, extra nitrogen) before adding, but I don't add directly so the worms can avoid until they want it. Used as a food, not a bath ;).

I'd also think twice about wood water.

I just toss torn-up corrugated cardboard on top, dry -- roughly a layer of food and then a layer of cardboard. It all breaks down in the end. How wet it's getting also gives me an easy visual check on overall moisture in the bins.