I've lived in Missouri so I know the kind of humidity you're talking about and the higher humidity in your area makes this an even better option for you
And yes you do get condensation at the AC unit (since the air is getting cooler in the AC you get condensation) but think about that for a minute.
The idea is to get the air in your case cool. You have a certain amount of moisture in your air (and yes you have much more in your air than I have here). If you were to just cool you case directly (either with water cooled or even a similar set up using actual refrigerant), and you were actually able to cool the inside of your case to below room temperature, that moisture would be forced out of the air and you'd get condensation.
But as you mentioned, the AC in your house drips water. That water doesn't just appear, the air is being cooled to a temp where it can't hold as much water and the water is being forced from the air.
Say the room your PC is in is 75-85deg with 90% relative humidity, and you hook it up so the air from your AC is ducted to your cases intake fan.
So as the air leaves the AC unit it is at 100% humidity for the temp it is at (but as far as actual amount of water in the air it is much less than in the same amount of room air, because air at that temp simply can't hold as much water).
But then it goes through the duct to your PC, picking up some heat from the rooms’ temp along the way. By the time it gets to your case it is much lower than 100%, and then in you case every single part (and any air in your case) is much hotter than this air so it starts to get warm. Hotter air can hold much more water, so the relative humidity drops even further.
Another example to consider is a de-humidifier. I'm not sure if you're familiar with how they work so I'll give you an over-view. Basically it's an AC unit that has the air path re-routed.
Your normal AC unit has 2 air paths, one takes air in from your room, passes it over the evaporator, sucking the heat out of the air, and collecting water as condensation.
The other is outside; it takes in outside air, passes it over the condenser the heat is driven from the coolant and picked up by the air, which is vented outside.
With a de-humidifier the compressor/condenser parts are the same but it simply takes air from the room, runs it through the evaporator, where it cools it, drives out the moisture (which is ran to a drain or collection pan), then instead of going back to the room it is passed to the condenser where it picks back up the heat and returns to the room at about the same temp it was taken in at but with most of the moisture removed.
By passing air through an AC and then directly into the heat of your case you are in essence making a de-humidifier. Not only will it not give you condensation problems or even increase the humidity in your case at all, it will have just the opposite effect of getting rid of any moisture problems you already have from your increased humidity (like when the increased humidity causes the dust to clump and pack itself in to places)
One more thing to consider before I get off this post that went on way longer than I intended.
Think about how dry your house get when you take cold air from outside and heat it up to 72-75 deg in the wintertime.
Cooling air causes increased relative humidity, and condensation. Cooling air and then heating it back up again greatly reduces relative humidity and makes condensation (outside of the actual ac unit) virtually impossible.
If the parts in your case are cold enough to pull condensation out of air directly from an AC unit than you have a case that is running well below 50 deg F and you don't really need to worry about cooling anything.
