You definitely are going to want to let it dry out a bit since it sounds like it's too saturated for your setup. I would see if you could find some kind of mesh bag. One big enough may be difficult but it would be the best way I can think of to keep bugs out but still breath very well and let the moisture out.
This is just an idea and may be a bad one lol, but if you'd like to use is sooner, you could probably put it in a pillow case, make sure it is securely tied closed and then throw it in the drier. Do this at your own risk and with some common sense like, don't leave your house while it's going, it may be a fire hazard so try at your own risk. But I don't see why it wouldn't work. It will dry it out nice and quick for you and then you can use it again starting with it dry and then ramping up the humidity until you get where you want. Just make sure not to use any thing chemical like softener in the dryer, make sure the pillow case is free of such things too. Well good luck.