Cannot move a project I created to a group library

If I understand your issue correctly, I think this may just be the way that Qualtrics has designed Projects and Groups to work.
It's a bit counterintuitive and less than optimal, but while you can drag or move projects between different folders within your own account, you cannot move the original version from your account to a Group. Instead, you can only copy it to share between your account to a Group, thereby creating multiple versions. Then, once the final edits to the project have been made, it has to be copied to the personal account of the user who will distribute it.
As an alternative to the process above, you can use the Collaborate feature to share the project directly from your account with other Qualtrics users so everyone can edit. That will save the work of copying the working draft into a Group and copying the final version back to an account.

Dear MatthewM
Thank you for your reply. It gives me a better idea of what to do with the group.
Kind regards