First, reasons for not using Multisite. And reasons to use it.
My guess is that you don’t need it, and that all can be contained inside a single WordPress.
Each Department would have a Custom Post Type. And the Organization keeps the default Post post type.
All of them share the same categories (news, events, others).
When showing a Department, the Query is done over the CPT and the desired categories. Same for the Organization.
Then, you’ll have to configure/adjust the theme and use the Template Hierarchy to display the main posts and the custom post types accordingly.
Case many Writers are involved, an extra touch is making custom roles and capabilities. So, each Writer of each Department can only access the correspondent CPT.
Plugins of interest: