Hello Dori,
If you are migrating the ArcGIS Enterprise from one machine to another with new machine name, you will have to install the software on the new server machine and then re-publish all the services.
There is no documented workflow to migrate services from one machine to another.
You can use the Enterprise Builder to build the new 10.5.1 version. All the software will be on the same machine. And when you upgrade the ArcGIS Enterprise to 10.6.1 you can use the Enterprise Builder 10.6.1 version to upgrade.