Soggy is not a problem.Recycling paper involves wetting it to loose the fibres and then reforming it. It's how paper is made.
But usually paper and cardboard that has been in contact with food is not recyclable because it contaminates the batch. That's why pizza boxes also cannot go into the cardboard/paper fraction.
> Soggy is not a problem.
It is when you're trying to suck a thick milkshake through one, though...