So, I have been creating a game on the downloadable version of quest. I would like to move it to the online version without having to publish it, this includes everything I've already done. Is there any way I can do this?

Upload to edit or to play?

You cannot upload it to edit. You can upload it to play it, and keep it as "unlisted". You can later upload a new version and you can also edit its listing on the web site so it is listed.