Reddit reviews Plastic Mesh Fencing 1m x 50m Green Barrier Netting Heavy Duty Weather Resistant - Ideal as Barrier for Crowd Control, Dog/Animal Fence

This is not anything I have personal experience with, but something I've seen done (in the UK, so also pretty wet)

Use some of the thinner logs to build a very basic frame, and anything you have to hand that will last long enough for a roof - I think the one I saw used leftover plastic sheeting, attached with a staple gun. The sides can be left open, but are best wrapped in something that will keep some rain out but let a draft through - the one I saw had a double layer of something like this. Pile up the rest of the wood underneath it, using the bark to cover the ground and the small branches stripped off the wood to keep the bottom layer of the ground and the rest of the layers separate when you stack it so that air can get between them.