I've had that happen before, it probably means your incubating temperature was a bit high but mine often hatch a couple of days early even under a broody hen. With mine I think it is heredity instead of incubating temperature. Still for next time calibrate your incubator before you start.
You do not want that wet chick to chill. You can put it in the brooder if the brooder is warmed up or leave it in the incubator. Try to keep it out of a breeze.
I don't know what kind of turner you have. As a minimum turn it off now. You don't want a chick to get trapped in it and hurt, a leg, wing, or neck. Can a chick get a head, leg, or wing caught in a sharp corner? If it can I'd want to remove the turner. Turners can be way different, I don't know which you have. With some turners, turning them off is all you need to do. The chicks will hatch even with the egg in the turner as you've seen.
Shrink wrap is a possibility but it doesn't really happen that often. I like to leave my incubator closed during hatch as much as I can to reduce that possibility but If I have an emergency I'll open the incubator to deal with it.
The eggs most at risk of shrink-wrapping is any that have external pipped. When you open the incubator have a spray bottle of warm water so you can mist eggs that have pipped. A light mist, don't soak the egg where it pipped enough to drown the chick.
Do not get in such a hurry that you drop eggs or have accidents. Don't waste time but don't rush. The eggs that have not external pipped aren't a huge worry.
Good luck with it. This is probably not nearly as bad as it seems right now.
