Basement floor is damp and the top of the foundation is cracking after air sealing the attic

Given all the experts that have looked at this first hand, I am hesitant to suggest a hypothesis based on other situations I have seen. Also I live in a dry climate, so forgive me if I miss something obvious to everyone who is actually there in New Orleans.

As you know, condensation is the result of warm humid air contacting a cooler surface which is below the dew point. From your description, you probably have slightly negative static pressure in your house and warm humid make-up air is being sucked through particular exhaust vents that connect to the outside. That the problem became less severe when additional dampers and insulation were installed supports this hypothesis. Unfortunately, short of a completely air-tight seal, warm humid air will continue to infiltrate through these paths as long as there is negative static pressure in the house. Even a slow rate of warm humid air infiltration will eventually make every cool surface in its path wet.

The central blower on a system naturally makes slightly negative pressure in a house due to imperfect ducts that leak in spaces that are not open to the return(s) of the system (under the house, for example). Any air that leaks out in such locations must be made up somewhere else, in this case by pulling a little outside air in through your bathroom exhausts. The same situation occurs if the structure is somewhat leaky and the bathroom exhaust ducts are relatively close to the system return. Return air flows via the path of least resistance and it may be easier to draw some air from a nearby bathroom exhaust than to draw 100% of the air from the far reaches of a house if remote supply air can more easily leak to the outdoors.

A relatively easy fix could be to give the return on your central air handler a controlled path for make-up air. For example, you could duct a small outside air inlet directly to the return of your central unit, adjacent to the main return with a balancing damper so you can make final adjustments. The outside air duct should probably be taped or sealed as well as insulated from the cooler temperature of the house. It would also need to be small enough not overwhelm the main return or affect your energy consumption too adversely. With a little make-up air provided directly to the unit from outside, your blower should stop sucking make-up air through your bathroom and dryer ducts.

You don't need a DIY forum, you need a lawyer or a politician. Your landlord is breaking the law. Good luck getting this resolved!