I've been using craft bales for a few years, it hasn't seemed to bother the cats' skin/fur at least; can't vouch for anything beyond that. I had the same concerns and spent a lot of time looking into it beforehand and went down a bit of a rabbit hole.

I'm not sure that the types of straw you find at craft stores are much different from each other, most of them specify that they're earth-friendly, biodegradable or sun-baked, and that the straw is highly flammable (it isn't treated to be fire resistant) so on the surface they do seem safe enough. Joann's Floracraft brand straw is listed on Amazon and they did respond to a question about whether it's safe for nesting chickens, they replied that their straw is all natural and didn't indicate there would be any other concerns (aside from removing the wires).

My only issue with the craft straw is that the pieces are chopped up quite short so they cling to the cat's fur and can potentially get ingested while they're grooming. For last winter I ordered an actual 4lb bale of straw off amazon to test out (they do have an 8lb option that's only $10 more if you have a lot of shelters or extra storage space). The straw pieces are full-length so it's easier to fluff up which provides better insulation, and the cats haven't managed to track it out of their shelters. My cats apparently prefer the old craft straw though so I ended up having to mix some of that back into all of my shelters to get them to use them again. Cats.