Provided both sites are at the same version of jD, and provided they have the same jD Categories and Downloads. Then it may be possible!
In your 'good' site save the contents of the jDownloads directory with say file zilla. This gets the categories and Downloadable files.
If you have your own images associated with Downloads then also save directory images/jdownloads/screenshots
At this juncture do a save in tools.
Make a backup of your new site just in case!!
On the new site replace the images/jdownloads/screenshotsand also replace the jDownloads directory, etc. Do not use jDownloads to view anything except Tools. And in Tools restore the original save
Should work but ...Colin