What is the best way to different department on a board but also have their own identity

You can add labels to cards (labels include text and color of your choice), so you can use those to identify which departments are working on which cards. Depending on how your organize your goal, you could make lists for each departments, and then specific cards for each task. If a member of the podcast team needs to help with a task on the writing team you can add individual members to cards. You can also make checklists on cards so that each department or person assigned to that checklist item can mark it as completed and everyone stays aware of what the checklist was and when it was completed.