I like extruded polystyrene foam boards attached with tapcon anchors and washers. You can use a construction adhesive with it but it's not needed. The foam has to be covered with 1/2" drywall for fire reasons. You can either apply drywall directly to foam or build a stud wall in front of foam.

Thanks. I want to avoid drywall, mainly because I just tore a bunch out due to the moisture problem and don't want to have to do that if the issue recurs.

You could use Rockboard.

https://www.rockwool.com/north-america/products-and-applications/products/rockboard/?selectedCat=rockboard%C2%AE%20downloads

Thanks @ksc36. I was only vaguely aware of rockwool and didn't know it came in rigid boards. That might be a contender. It sounds like expense and friability are the two main downsides to it.

There's also Thermax, though I'm not familiar with it. Looks like it's about $1/sf for 1", R-6.5.

https://www.dupont.com/products/thermax-sheathing.html