@bl4s7er, @Anonymous , @M0J01 and the others affected by this: are you member of a team inside Fusion 360? I ran into the same issue you have and did a lot of reading until I finally found the solution:
https://knowledge.autodesk.com/support/fusion-360/troubleshooting/caas/sfdcarticles/sfdcarticles/How...
My account was created way before the end of June 2019 and so I did not have a team hub only my single-user storage containing all the projects I created throughout the years. Once I followed the instructions provided above and switched to the "team" I created, I could create new projects again.
From my understanding every user needs a "team" from now on, even if you're only doing personal projects irrespective of the type of license you have(?). Looks like the requirement may have been introduced in 2019 but didn't affect many old users and only the recent changes highlighted the issue!? Maybe one of the moderators here could clarify whether my assumption is correct.
I just wish it were made more obvious for old account owners that they have to create a team. I bet a lot of the issues with new project creation are caused by this.