Peltier junctions produce a temperature difference between one side and the other. This temperature difference is inversely proportional to how much heat (in watts) it is pumping out of the cold side into the hot side. You've already seen that this happens very readily. If you have no fans or heatsink on the cold side, then very little heat circulates into the cold side, and thus the TECs don't move much heat. They don't move much heat, so they produce a large temperature difference.
You mention the temperature outside of the suitcase, but that information is totally irrelevant to any of this. What does matter is the temperature of the hot side of the TECs themselves. I don't mean the temperature of the heatsink mounted to them, I mean the temperature directly at the hot side.
This is the temperature they cool with respect to. If they are 60 degrees C on the hot side, and 40 degrees C on the cold side, then it is working and cooling. Sure, both sides are above room temperature, but that doesn't matter. TECs only create a temperature difference between their hot and cold side. They don't care what the temperature in a room is, or what you think hot and cold is. Considering that those TECs are generating over 100W each of waste heat, and after you've added heatsinks and fans to the cold side which will greatly increase the the wattage they pull out of the cold side, each TEC is shedding 120-150W on its hot side.
Let's be extremely generous, and assume your hot side heatsinks and cooling is just amazing. These things are soldered using bismuth-tin low temperature solder directly to an array of heat pipes which, at the fair end, are soldered to a massive radiator, all solid copper, with a 140mm fan going like a jet engine, literally hemorrhaging watts into the air. The cooling is so fantastic that the total thermal resistance of the hot side plate to ambient is only 0.3°C/W. This is just shy of twice as good as the 140W 14 core Xeon workstation class CPU heat pipe radiator coolers. And, thanks to the fans and heatsinks on the cool side inside the suitcase, you've made it so quite a number of watts are able to get sucked out by the TECs. The maximum your specific modules can do is 75W. Let's be conservative, and say that the cool side heatsinks are small, fans slow, and thermal interface poor, and they're only able to move 30W of heat each.
Assuming you're running them full tilt at 6A, and moving 30W, you'll generate 104W of waste heat and an additional 30W from the heat which is actually being moved. This is amazing by the way - TECs typcially manage to move only 1W of heat for every 5W of waste heat they generate.
Anyway, moving 30W, the datasheet says they will produce a temperature difference of about 35-37 degrees C. And since your cooling of the hot side is absolutely amazing and could keep a 14 core Xeon monster significantly cooler than Intel's own heatsinks that come with such CPUs, that 134W of heat the TEC is dumping out of its hot side will only raise its temperature 40°C.
Not to 40°C. By 40°C. If room temperature is 34.9°C, then your best-in-class cooling of the hot side is (impressively) able to keep the hot side of the Peltier at the temperature of about 75°C. This means the cold side of your TECs will be 38° C, assuming 37°C of temperature difference. These TECs are working great and cooling a very good 30W (10-20W is a much more realistic figure, 30W is basically wishful thinking). Sure, both sides are still warmer than room temperature, but what did you expect? You're dumping 100W into it. Of course it's above room temperature. Thanks to the TEC doing it's job, and doing it very very well, one side is only slightly above room temperature.
So, based on what you've said, it sounds like your TECs are working great. The only real problem is, unfortunately, your expectations. There is nothing wrong with the TECs.
If you actually want to cool something below room temperature, that means keeping the hot side near room temperature as well. You do that by limiting how much heat you are actually pumping with each TEC, run them at a fraction of their rated power, and have the most ridiculously high performance hot side cooling as you can. The easiest way to do this is just reduce the number of watts you need to dissipate. Your cooling is adequate to run them continuously but only if they have to remove much less heat.
However, there was so little heat being moved that the temperature difference became too large and they began to condense humidity out of the air. Condensing moisture out of the air doesn't just happen as a side effect - producing a phase change in material requires overcoming it's latent heat. This is the same thing that keeps a pot of boiling water at essentially the same temperature. All the heat energy is going into turning water into steam, and none goes into actually heating it up more.
If your TECs are condensing water, then all of the wattage they are pumping out is being used to turn water vapor into liquid water and none of it is going to actually go into cooling the humid air. You want to prevent the cold side of the TECs from getting colder than the dew point, or make the suitcase air tight (which it should be anyway if you expect to actually keep the inside cool) so the TECs can completely dehumidify the air inside the suitcase, at which point they will finally begin cooling the air.
Try lowering the power, and use aluminum plates on the cold side rather than heatsinks and fans. You want a heat spreader on the cold side, just something to boost the surface area and thermal mass some. With the heatsinks and fans, the amount of heat being moved is to much that the efficiency has fallen to the point that the TECs are basically just cooling their own heat out of themselves and little else.
And, of course, take whatever your cooling strategy is on the hot side, make it twice as good while cutting your expectations in half for performance, and then I think you will finally get results that you want.