Changing LAN Scope

I need some advice! I have inherited a LAN that is using a public IP Scope and need to change it to a private scope – i.e, 192.168.0.1 through 192.168.3.254. We are a 24/7 healthcare center so I need to be at nearly zero down time. I’m thinking of adding the private addresses in a superscope and meticulously moving our fixed IP’s – printers, dication systems, radiology systems etc. to the new scope and then removing the old scope. Anyone have any ideas how to proceed with this?

Any help or expert advice would be greatly appreciated. We are currently have IP conflict issues because our PC’s are receiving IP assignments for an external DHCP server as well as our internal one.