There's no inbuilt functionality in Azure Devops to move the Organization and Entire Project to new Organization or target organization. If you don't want to use third party tools you can move the work items, Pipeline, History, Repos by creating a script and using Azure Devops Rest API's.
You can create an automated script and call Azure Devops Rest API and Fetch the WorkItems, Pipelines, Repos with Get Request and call the Post API to Post the output payload from Get request in the Target Organization or Project to create the same set up.
Refer this links for more details on the same: a) https://learn.microsoft.com/en-us/rest/api/azure/devops/?view=azure-devops-rest-7.2 > in the left pane check rest api's for different Azure Devops Services
c) https://www.baeldung.com/ops/azure-devops-migrating-project-different-organization
d) https://stackoverflow.com/questions/74793329/how-can-i-export-a-list-of-pipelines-from-azure-devops
Refer my answer here for more details: https://learn.microsoft.com/en-us/answers/questions/5630894/how-to-move-organization-for-azure-devops-service