Moving to the new server is much safer since you don’t need to change the old server until the new one is working!
What I would do is follow the rsync guide, excluding postgres_*. Then I’d rename app.yml and run ./discourse-setup --skip-connection-test (because the DNS won’t be pointing to the new server yet. Then restore the backup. You can (usually) test that the new server works by changing your local DNS to point to it, but worst case, you just change the DNS and if it’s a disaster you change the DNS back. (If it’s digital ocean or something else with a floating-ip that can be pointed to multiple VMs then you can just re-point than and not fuss with DNS.
I’ve done this about 10 times in the past couple of months. If you’d rather not do it yourself, I’m available.