*Marking this post as solution for visibility because this is a long thread - not considering this issue fixed*
Thanks to everyone who answered my questions to help us investigate this problem. We do not need more information at the moment, but again, thanks for taking the time to explain what you did in your games before it stopped saving!
For those of you who are wondering, in case you missed my previous posts - yes, the team is working on this, that's why I asked you for information 😉
In the meantime, alecianfli226's workaround seems to be helping most of you: https://answers.ea.com/t5/Technical-Issues-Consoles/Sims-4-won-t-save-on-Xbox-and-PlayStation/m-p/6427127/highlight/true#M134
Edited: @helen26686 has also shared a workaround that helped them: https://answers.ea.com/t5/Technical-Issues-Consoles/Sims-4-won-t-save-on-Xbox-and-PlayStation/m-p/6444552/highlight/true#M611
Remember to give them some XP if it's helpful for you!
Thanks for you patience,
-Mai