Is there a setting, or a third party application that will allow me to do this? My school gave me unlimited Google Drive storage, and I want to take advantage of it by backing up everything.
To get this result, put the file into a folder within GDrive, wait for it to sync to the server, then set your desktop GDrive settings so that folder is no longer sync'd to your local disk. This will keep the copy on the server and remove the desktop copy.
Once you have GDrive folder(s) that are not sync'd, you can upload more files into it/them via the web page uploader. Alternatively, put files in a sync'd transfer folder, let them sync, then use GDrive's web interface to move them into an unsync'd folder.
Google got another app called "Drive File Stream" for that use and business https://support.google.com/a/answer/7491144?hl=en