railway doesn't provide temporary access for expired trials to backup your database, you need to upgrade to the hobby or pro plan once upgraded, redeploy your mysql service and your data will still be there.

then you can export it using mysqldump -h switchyard.proxy.rlwy.net -P 55110 -u root -p railway > backup.sql or any mysql client like MySQL Workbench or DBeaver, you have 30 days from trial expiration before Railway permanently deletes the data, so act quickly!