We had to move out of a rental house last February because of mold. We lost so, so, many possessions because of it. All of our beds and mattresses, our couch, two sets of bookshelves, not to mention all the small things. It was heartbreaking. Our lease was up, and the owners (a foundation) didn't want to pay to replace ANY of our belongings. For MONTHS we had been begging them to fix the problem, sending them detailed pictures and letters, but they refused. We had to threaten to sue for them to give us a few thousand dollars, which didn't even begin to cover our losses, but it was something, at least.
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Don't even bother trying to remove the moisture until the problem is fixed. Our problem was a mixture of leaks and a gas furnace that put too much moisture in the air. We ran a dehumidifier 24/7 and all it did was make our electricity bill skyrocket.
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Document everything. It's only a rental, so I'd get out of it. The house is unsafe and damaging your possessions, it should not be too difficult to get out of your lease. Get out as soon as you can, before you lose more of your things.
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And when you move, you want to be sure not to introduce mold into your new house. We washed every article of clothing, towel, sheet, etc, and then bagged it up and moved it out of the house as soon as it was out of the dryer. (I started this weeks before we moved, it took forever.) We wiped down all hard-surface furniture and toys with clorox wipes. I washed all our dishes (the mold was even in our cabinets) in hot water, dried them, and boxed them up and moved them out.
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Anything with mold we left behind, particularly soft furniture that wasn't possible to wash, ie, mattresses. I did NOT want to have to deal with mold all over again.
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Good luck to you. I hope you can get out soon.