I just went through the exact process yesterday. 
Create a blank course on the new server and note the course number. I named it Transfer for simplicity. Make sure it has enough weeks or topics to match or exceed the largest course you are moving. Back it up. That created the backup folder.
Download all of the zipped backups of the courses you want to move to your computer.
FTP to the new server and locate the uploaddatta folder. Locate the folder number for your the course created. Go to the backup folder and upload all of your zip files.
Now login to the new server, go to the new course and restore the courses one by one as new courses.
You can delete the Transfer class when done.