add a sleep before your app starts by changing your start command to:

sleep 3 && node server.js

(or whatever your start command is)

railway's private networking needs a few seconds to initialize. if that doesn't work, check if you're using alpine linux - you'd need to add ENABLE_ALPINE_PRIVATE_NETWORKING=true as an env variable to your app service (not the database).

also make sure you're using DATABASE_URL and not DATABASE_PUBLIC_URL for the private connection.

check your schema and root passwords are correct