How to create a new project without closing the one that's currently open?

Jeroen,

Do not export from SL. If you attempt that, as you say, the only way is to publish it first, and you cannot edit a published version.

DO NOT copy one slide and paste into another project, that will likely introduce corruptions that will hide your true problem when you post it on the forum.

Open a new project in SL. Yes, it will close the first project (which is probably not a bug, but a safeguard against corruption, given the way SL saves a project). In the new project, IMPORT a slide from Storyline. Navigate to the project in question, choose the slide you want to import. You will have a .story file with the one slide you want, suitable for posting on the forum.

Now you know a second, safer, much better way to move material from one project to another.