Hi
As soon as you duplicate a project, it makes a copy of all the tasks belonging to the project.
What you could do is saving our project as a template (before adding tasks) so that you have sections and custom fields already sat up when you create a new project. And then you multi-home the tasks that should be in the agenda. Multi-homing doesn’t duplicate a task.
When duplicating a project, you can unselect the “assignee” but I don’t advise you to do that because you’re ending with 2 tasks duplicated for the same action.
Or you can have a look at the project I use with my team, where columns are weeks so that I don’t have to create a new project every week, this is also an idea: 🏆 The Asana coolest projects Award - #23 by Bastien_Siebman