Sabrina, you wouldn't necessarily see broken roots, firstly being in the air during transfer, unless it was very quick, can dry out the little hairs on the roots, and secondly when you firm the soil around the seedlings it can further damage the roots. Unless you plant with a rootball adhering, you're inevitably going to have done some damage to the roots.
Are you sure they don't need watering? Cutting the leaves back would be the best thing, and shade either by my method or Arlene's. You're more likely to lose them if you don't cut the leaves back.
