Right - Asana does not shift tasks in an existing project. The recently-released feature that’s the focus of this forum thread only shifts task dates when you first create a new project from a template. So what you’re mentioning here will not cause dates to shift.
When you say you set the new dates: did you do it specifically on this “Step 2 of 2” screen during the project creation process?
That screen is the only place where entering a start or end date will cause tasks to shift - and you can only enter one or the other (start or end), not both. (You mention setting both start and end dates so I’m thinking this is not the screen you used.)