The basement has a little moisture, but nothing a dehumidifier won't cure.  

Moisture helps mold to grow so purchasing a dehumidifier would help remove moisture.Good luck

In Texas the ground bulges or cracks depending on moisture or lack there of.

But let's say a seller says, "oh that, it's really nothing, been that way for a couple years, my Realtor put that in because, technically some moisture comes in under the roof and has caused a stain on the bottom of the roof decking.

The entrance to this unit faces out from the rear of the house.There is an issue with excess moisture in this unit in particular.

and once the bank took it over after buying it back at sheriffs auction, someone came and secured the house and made sure every last window was tightly shut and sealed and this was late spring, so the house sat all sealed up through the summer and a teeny tiny bit of mold popped up in the master bath because those casement windows were rather beaten up and didn't seal 100%, a just a tiny amount of moisture got inside and made the tiny bit of mold and the REO agent who was listing the house mentioned it in the listing, but he put something like "NOTE-mold issue is master bathroom!!"

I would however pour in some waterproofing sealant on the floor to prevent upwelling of moisture. 

What products/methods have people used to help "soak up" dampness and moisture in a basement?

There is no moisture now in the basement I just don't nail trough the floor concrete.  

I figured the moisture was a good part of the reason for the roaches.