Winters in Tulsa are quite cold, especially with the humidity.

Another issue is they use fan and turn it off right after they are done and bathroom is still humid and steamy3. 

Primer doesn't hold up to water / air born vaper / steam / and humidity.

a couple of box fans + 20x20 filters.... run continuously 1-2 days. dump the filters directly into a trash bag. replace and re-run.have the ac/heat ducts (if any) professionally cleaned... you don't want to breath the nasties again....also i would check for cold/humid sections of walls. also crumbly (drywall/plaster) around baseboards. if you have those, (respectfully sorry) you are in for a surprise if you open the walls.....now, what is the source of humidity?

Call another company out to do a bid and get someone in that knows how to properly check the humidity level in the house.. 

Summer temps in the triple digits in 80+% humidity are the norm not the exception.I wouldn't rent a property out in SC without A/C ...that's a recipe for a lawsuit.

Mold is caused by water and humidity. 

Just like Southern California, only new and improved with 50% more humidity :)

Condensation is simply a function of dew point- when air of a given relative humidity contacts material of a given temperature, condensation WILL occur.

The basement now has sort of a stale, humid smell.