A lot of it is used to make paper which degrades relatively fast. That being said it's still renewable, fairly recycable, and provides packaging that actually degrades instead of clogging the ocean, so we don't really want to mess around with that side of the equation if we can help it

A lot of paper in the US ends up in landfills. Paper in a landfill is sequestered carbon, since it basically does not decompose once it’s buried.

So, if you subscribe to a dead-tree newspaper, it might be better for the atmosphere to throw away the paper instead of recycling it.

Indeed, logging is a measurable carbon sink. It's not a very good one though, as really we're harvesting about as many trees for lumber as we can turn into houses locally.

My understanding is that a lot of the carbon is still released as the sap is dried out and so on.

No, that's not happening. The carbon is by and large stored in the cellulose, hemicellulose and ligning of the wood.

That carbon will go back to co2 again if the wood rots or burns - but not when it dries.

Got any source on that? Carbon generally isn't released without burning or so I thought. Though, this article seems to suggest that composted wood releases it's carbon which is at odds with what I thought I knew.

Biological process are generally a slow controlled burn. Compost burns the carbon for the energy that creates, releasing CO2.