KMS2222 the instructions can be found at https://techcommunity.microsoft.com/t5/azure-sql/using-failover-groups-with-private-link-for-azure-sql-database/ba-p/1272902
But considering what you said the problem could be related to DNS Name and TTL (Time to Live)... When you get a DNS registry it will be valid for usually 1h. If client is a VM you could do a IPCONFIG /flushdns... If webapp, a service restart probably may do the trick, or just need to wait for the TTL.
You can validate it using Kudu on WebApp and trying to check name resolution NSLOOKUP failoverserver.database.windows.net. It should point first to one region gateway, then to other one. Here is a sample of a NSLOOKUP I done on my failover name... You can see that currently lead to West Eu
nslookup failover.database.windows.net
Non-authoritative answer:
Name: westeurope1-a.control.database.windows.net
Address: 40.68.37.158
Aliases: failover.database.windows.net
primary.database.windows.net
primary.privatelink.database.windows.net