New Ahrefs v3 API: We will be upgrading to the new Ahrefs API on 18th October. All customers will need to re-authenticate the Ahrefs integration after the upgrade. Here is a blog post on the migration
To find available domains you can select the "Available" checkbox in the Advanced Search box. Note that the Archived Section only contains available domains and therefore the "Available" option does not appear when you search in it.