How to get Birth Certificate

Michigan does not have any provisions for readoption. We went through our county court to change our children's first names, and as part of that process their Michigan Delayed Registration of Foreign Birth was requested from Vital Records in their new name. We received their new BC's a couple weeks later. I found Michigan Vital Records to be no help at all. They just told me that I had to go through my county court. Honestly, I would start there.