Electrical – Replacing bathroom exhaust fan without ground

There is nothing in the NEC that prohibits the installation of surface raceway (metal or nonmetallic) in even a damp or a wet location provided the conductors inside are rated for said damp or wet location.

That said, many AHJs do not consider the area outside of the shower zone to be a damp location to begin with. So, go ahead with the raceway -- just remember to use a wire (such as THWN) suitable for a damp or wet location when running wires inside the raceway.

As Ed said in his comment, you need to reconnect your exhaust to the roof vent. Perhaps your question should've addressed that issue. It's not clear why you made that move.

I also agree that the insulation will probably dry out well with a little air movement. I'd place a fan blowing across the surface of the damp area, and be sure that you have good outflow. Apply a fan to a roof vent for a few days if necessary.

The drywall isn't an issue either, unless you have visible signs of water damage or mold. Even mold on the attic side wouldn't motivate me to tear it out.