If you don't solve the humidity problem you won't solve the bug problem. I hope you have a kitchen extractor fan. So you can keep that running anything you cook (especially boil or steam).
Also, you can wash the black mold in the bathroom and walls with a baking soda solution. You don't need to use expensive cleaners.
Otherwise, personally, I would also look for some place else if at all possible Humidity is difficult to take care of and not good for your health. If the landlord isn't willing to deal with the cause of the problem you are fighting a losing battle.