It is very easy to re-use a learning activity which you have already set up. First, keep in mind the name of the instance which you want to copy. Then, create a new, blank module of the relevant tool in the same or even another course. Access it, press "Get Started" and you will be be met with the usual set up screen.
In set-up mode, you can access a small drop-down menu by clicking on "Use copy of existing", just above the first step.
The dropdown menu will contain the name of the old learning activities you've set up in the respective tool along with their date and the name of the creator. Select the one you want in this list.
As soon as you click the desired version, all associated settings, as well as the file the respective version used, will be copied from the old version into the new one.
Once you will click "Save" you will conclude the process of copying an old assignment into a new one. Make sure, however, to set a fitting name for your new learning activity within the LMS interface also.