Time management for managers

I re-joined scholars specifically for the time management month. Here’s my situation. I manage a labor and delivery and high risk OB unit in a hospital. My scope of responsibility is broad and in our department we have 5 different types of service/care within one unit. I have 65+ direct reports and 24/7 responsibility to the management of the unit. I feel like there is no such thing as a typical day and struggle to keep up with what has to be done and the projects and initiatives that we are moving forward – the result being 12 hour days, 5 days a week with catching up on the weekend. Even with scheduling, I could be called into an emergency at any time and would have to drop what I’m doing for anywhere between 10 minutes to the entire day. Recently, a few things have escaped my notice and I have also missed a couple deadlines of a more personal nature that also have some carry-over to work (certifications, for example), which is just not like me – I’m usually an over-achiever and somewhat perfectionist.
I have some ideas on how to manage my time, but keep stumbling against the sheer volume of work and unavoidable interruptions.
I thought that maybe I’d schedule in a couple hours of “putting out fires/emergencies” but then think that if I didn’t have an emergency/situation I may not use that time productively.
When I do have “down-time” I am exhausted and barely leave the house.