Ya, as soon as the roots come out of the
growing medium they will die unless they are continuously misted (a
aeroponic system). Even in a aeroponic system where the roots hang in mid air and are subject to high humidity levels, the roots still need to be misted way more often than flood and drain system cycles. Generally aeroponic systems cycle something like 1 minute or less on, and 5 minutes or less off. What you need to do is get the roots to grow long enough (to the bottom of the NFT tube) before you can run it as a NFT system. That's why I recommended raising the water level in the NFT tubes (as a flood and drain system) to keep the roots wet until they reach the bottom of the NFT channel. Dripping the water down from dippers can work for this purpose as well, by keeping them wet as the water drips down the roots on its way to the bottom of the NFT channel. However dripping it down from above may take more trial and error to get just right, as well as less complicated than starting it as a flood and drain setup. So running it as a flood and drain system is simplest and cheapest to me. Either that, or build a separate system (like a simple water culture system) to keep them in until the roots are long enough.
Also If I'm not mistaken your using the polyester material as a growing medium, and from what it sounds like the polyester material just dosen't hold moisture very well (but actually rather poorly). That would be contributing to the problem because it should be able to hold moisture when suspended in the air. If not it would need constant moisture from some source to keep the main root-ball wet. Thus rather than being able to slowly drip moisture down the roots as they grow down (from the growing medium), it rapidly drips all the moisture out quickly. Leaving the main root-ball to dry out, and nothing to drip down to keep the roots searching for the moisture at the bottom of the NFT channel wet until they get there.
P.S.
I would bet if you simply filled a couple of baskets with coco chips/fiber and stuck your plants in it (even with the polyester material), and changed nothing else (assuming your spraying the bottom of the baskets with a 15 or 30 minute on/off cycle), the plants in the coco chips would grow roots out of the growing medium, and all the way down to the bottom of the NFT tube.