To make separate tasks, you should Duplicate the task (one way is to right-click and choose “Duplicate task”).
You would not want to add an existing task to a new project because that would create two references in different places to the very same task, which doesn’t sound like what you want.
And that may be the source of your problem: Perhaps the task you’re duplicating is now a member of two projects causing the problem.
You want to 1) Duplicate task, then 2) probably make the task only be a member of the new project.
More info is here: A task’s associated projects.