Update!
I started ripping into the floor today, and was delighted to find a solid not so rotten wood frame! The rotten floor is just plywood nailed to the frame. So I'm going to work at removing all the plywood and then decide which direction to go in after that. The only thing is the shed was nailed to the plywood, so when I remove the plywood, there's a gap between the shed and the frame

The edge of the shed is nailed to the frame but because of that floor there's that gap. We will probably end up having to pick the shed up off the frame anyway.
Now I need to figure out why the floor rotted in the first place. The dirt is pretty wet under it, I worry there will be too much moisture in the coop.