Anybody know about this old building material? [health concerns]

I rented a place site unseen and there is an extremely strong dust/mildew/mold smell. The worst part is in the closet. It's an old cardboard type material that in squares about 16" x 40" or so. It smells awful. Family came down and they would not stay here. I want to turn the room into an office, but the smell is overwhelming. It makes my lungs hurt.

I am concerned it may be causing health problems. I mentioned it to the landlord and he pretended not to smell it. When I put on an N-95 mask the smell goes away. I am not sure if it's annoying or a serious health concern. I am having trouble finding information about the material. It smells awful and I've finally narrowed it down to these old 1930s cardboard walls.

Here are links to pics;

This one is in the closet and is the worst part. I primed it with mold killer paint and it helped but is coming back:
https://photos.app.goo.gl/NEBCq5fTQfQCiqzC9
https://photos.app.goo.gl/c4rsqxg7BE88KEmc9

And it's on the ceiling:
https://photos.app.goo.gl/mrkAPUPdfkB4vzhL9

This is where it appears to be exposed in the basement:
https://photos.app.goo.gl/PXU4iseSQxyZuuM58

Please, let me know if you have any ideas about what it my be. It's a 1930s Colorado home (early New Deal) and the guy built it himself. It clearly did not use the best building materials, but I am concerned it may be a health problem. I hope it's just a health issue.

Thanks for any help