While roaches can survive with very little water and on any food source they would probably appreciate more than what a carrot can provide. Some people use potatoes or cucumbers for insects that would drown in a water dish or you can
Cricket Quencher which is gel that holds many times its weight in water. Another alternative is to soak cotton balls in water and put them in a dish but this requires a lot of attention as it dries out fast and can get messy so the cotton needs to be changed regularly. There are a variety of roach diets on the market or you can find a recipe to make one at home. I make my own from baby cereal, fish food, bee pollen, brewer's yeast, and bran flakes. Since you are feeding these roaches to your mantis you want the roaches to be well fed and healthy so your mantis can benefit from their nutrients. There is something called "gut loading" where you feed the prey insect an extremely nutritious diet that is high in vitamins so that the mantis gets the benefit of whatever is leftover in their digestive tract.