Similar to Audit Logs, CircleCI retains build logs for up to one year based on the repository name. As a result, logging back in and re-setting up a project will retain the logs. In order to keep logs for both projects, we recommend you increment your repository name, i.e. project to project-1 or in a way that best suits your needs.
If you would prefer to delete projects on your own rather than reaching out to Support, you can vote and comment and vote on an open feature request to support project deletion in the UI.
Note: Full project deletion is permanent and logs are unrecoverable once your request is complete.