Natural dye for hand spun fibre - before or after spinning?

So… I was out on my traditional noon time walk down a local hike and bike trail and noticed a group of prickly pare cacti that were absolutely covered with cochineal webs.

Geek/naturalist that I am my first thought was something like: “Hey, I should collect a few of those and try my had at traditional dying”.

I count spinning/weaving among my arcane interests and have been making some wonderful yarn from the undercoat of my golden retriever/shepherd mix. (don’t laugh, most folks pay enormous amounts of money for sweaters made from the hair of an animal that spends most of it’s time walking around with it’s own poop stuck to it’s butt, whereas my dog is cleaner than most people’s kids.)

My question is, should I dye the fibers prior to carding and spinning or should I dye the skeins after they are already spun?

As a side question. Do I need to remove the cochineal bugs from their webs or can I just grind them up web and all?