Giving them different categories is ok – it would save you some query modifications as when user enters a category page, only events belonging to that category will be displayed.
The full calendar display is also customizable, but in a more complex way.
Basically there’s a shortcode for this: [em-full-calendar] but unfortunatelly it does not allow you to exclude / include categories, yet. It does have multiple options but not for categories – I just added this to EM todo list.
For now you can use the em_get_events filter to modify the events query in calendar on selected pages using wordpress conditional tags.