My strategy is to setup the new server, and once everything is ready, stop the service on the old server and replace it with a reverse proxy to the new server,

My strategy is to setup the new server, and once everything is ready, stop the service on the old server and replace it with a reverse proxy to the new server, and THEN finally update the dns record. When the old server no longer see any meaningful traffic, I'll decommission the old server. This way I don't have to deal with both server out of sync.