You need at least version 5.10.4. If that's not the case update your v5.
In v5 run a database integrity check in Acymailing->Configuration->Security. Then export your v5 tables using phpadmin or similar and import these in your new site. You can safely do this, both versions use their own set of database tables. The ones starting with #acymailing refer to v5 and the ones starting with #acym refer to v7.
Once you have imported these tables go to Acymailing->Configuration->Security and start the migration. https://docs.acymailing.com/setup/migration