My personal experience is that the window units are more efficient - maybe its something with the deep south 94-99% humidity and subtropic temps? 

I would assume that mold tends to be a concern as well given the high humidity levels in Florida.

The foundation isn’t cracked but rather the sewer pipe coming through the foundation was not sealed properly.We were thinking of putting cement around it and resealing with tar.The joists were never underwater but wet from the humidity and moisture.

And we didn't even have fans, just stale, humid, non-moving 100 degree air. 

With this, fresh airflow is very important to avoid stagnant air and humidity within the home.

They determine the mold is due to; (1) tenants causing crazy high humidity basement humidity by running the dryer without ever cleaning out the lint trap and (2) not cleaning the bathroom- All mold is abated with bleach, a second dehumidifier is placed in the basement, carpets are lifted, trim is replaced, and we investigated behind walls.

I am afraid of the humidity though.. 

Houston is getting Hot and humid slow and steady.

We don't have any AC in the house yet (it has been hot/humid....but so has the other side of the duplex where the patterns are not showing up).

With no A/C on, in that heat and humidity fleas can be a huge problem.