I would try to keep employee only messaging off Facebook – it could confuse the audience and make followers less engaged/unfollow the page. Posts promoting the company culture, employee event recaps, etc. are one thing but as you said, a public facing account is not the right place for employee only announcements.
At our network, we have a specific "Employees Only" webpage that houses all announcements, quick links to commonly used information & logins, HR information, etc. It is only accessible while you are in the hospital wi-fi network/on a campus. Would something like that be an option if you don't have one already?
My other thought would be network wide email blasts or a weekly/bi-weekly/monthly email created as an 'employee newsletter' with important updates – this would ensure ALL employees are receiving the messaging & HR can check open rates. This would work better than social media because you wouldn't be able to track how many employees specifically see the message. Hope this helps!