Mildewed carpeting, uninsulated flooring
The problems you describe are likely on the outside of the building. I'm willing to bet the landlord is unaware of this, or they are simply unwilling to fix it. This likely isn't fixable by you.

The moisture is coming through the concrete. Concrete is porous, so it will wick water from the surrounding soil naturally. To help prevent this process, basements are typically surrounded by a weeping tile. This is how the outside of the building basement should work



In new construction, this is corrugated pipe with a sleeve to minimize dirt, but in older buildings this was literally clay tile. So, what happens over time is this tile can break (it's ceramic), blocking the weepers. There's no way to find this out, except to excavate the outside wall and look at it. (image source)



My bet is


The weepers are broken
The wall has no waterproofing
The ground does not slope away from the building (you can confirm this visually)


Repairing this is not cheap, obviously. It's not surprising that it was ignored if it was known about.

If the joists are spongy, it means they've absorbed a lot of water. I'm surprised it hasn't attracted termites yet. This could be making the apartment structurally unsound if it has been an ongoing problem. Even if there are no termites, the wood may begin to rot.

The only thing you can try yourself is a dehumidifier. But if the room is humid enough to wet the floors and joists, a dehumidifier will likely be overwhelmed very quickly (you will have to empty the collected water frequently). The owner should call in a foundation waterproofing company to excavate the basement walls, water proof them, install new weeping tiles, and slope the ground away from the structure.