quomii I have privacy concerns such as people knowing my sexuality, politics, religion, etc.
There is a fair chance Facebook (for example) already knows all of that, plus probably a couple of the ad-serving networks.
quomii I feel like Apple Intelligence will make this information less than anonymous.
Maybe? They have at least invested in technical infrastructure for your device being able to do off-device private computing, i.e., actions rather than just words (source).
quomii Should I get a new SIM card and a new number?
It's not clear to me how a new SIM card would help. It wouldn't help with location tracking. It also wouldn't happen with behavior tracking. If, hypothetically, you have a Snapchat account, and you were to destroy it and create a new one on a new device with a new SIM card, if you then rejoined your old groups and the "new you" connected to your existing friends and coreligionists, probably you would have spent a lot of time and effort only to end up where you started.
The problem is that the things that are straightforward and obvious (new SIM card) don't address the core of the problem (location tracking, behavior tracking). If you talk to your friends (etc.) using privacy-focused end-to-end encrypted networks, you can be tracked less. But that requires convincing your friends (etc.) to change their habits too. The problem is a hard problem, so steps that are simple and easy probably won't get you very far and might not get you anywhere. To get somewhere substantial you will likely need to behave differently, permanently, in ways that will disrupt some of your existing habits, because your existing habits were steered in part so people could make money by tracking you.