Can you give some details or a picture of their set-up, so we can see what you see? They do need heat usually until they are fully feathered and around 6 weeks old. Are they in a brooder, and how are you giving water? Chick waterers should be set around shoulder height and kept full to keep them from turning over and spilling. I prefer a gallon plastic waterer for that. Some use nipple waterers, but I would stick with a standard waterer in freezing weather.
Whenever chicks get wet they need to be dried thoroughly so they don’t get hypothermia. Coughing can be a sign of a respiratory disease. Wet conditions in a coop can be a good place for mold to develop and aspergillosis is a respiratory infection from mold. Other common respiratory diseases in chicks are infectious bronchitis or mycoplasma (MG.) So I would get things dry, change the bedding, raise the waterers on 2 inch boards or a platform, and provide heat to prevent chilling until they are feathered.
