Problem: New MacBook Air Tahoe 26.2 "Update Apple Account Settings"—wheel just spins. On iPhone changed password—still spinning.
On old MacBook Air [Monterey 12.7.6] could update PW on my account.
New MBA and Old MBA using WiFi could screen talk together, not now.
MacBook Air 13″, macOS 12.7
Since the password works on your older Mac, I wouldn’t keep changing it. The spinning “Update Apple Account Settings” message on the new Mac can also happen when it can’t complete the connection to Apple’s account services.
I’d first restart the new Mac, make sure macOS is fully updated, and check Apple’s System Status page. Apple specifically recommends checking for an outage when Apple Account sign-in just keeps spinning.
Also check System Settings > [your name] > Sign-In & Security and see whether there’s a specific item still asking to be updated.
I wouldn’t sign out of iCloud yet, especially on a new Mac that may still be syncing your data.
Apple’s current sign-in guidance is here: Sign in to your Apple Account on the web or in your device settings - Apple Support
New MacBook Air 'Update Apple Account Settings' wheel spins
