Xbox account sign out and sign in issue

I understand how frustrating this is, Adam Watson.

I've came across a similar case and the cause turned out to be Xbox Game Bar still being signed into that account. Though not identical to your issue, I think their suggestion would be worth a shot. Please try this:

1. Sign out of the Xbox app > immediately press Win + G to open Xbox Game Bar.
2. Click the gear icon > see if there's Accounts/Profile section > sign out as well.
3. Close both the Xbox app and Xbox Game Bar > open Settings > Accounts:
   1. Check under Your info / Email & accounts (also Accounts used by other apps)
   2. If you don't use that account for anything, Remove it from the sections above.
4. Open Microsoft Store > if the account is also signed in there, sign out > restart your PC.
5. Open Microsoft Store first and sign in with the Microsoft/Xbox account you actually want to use.
6. Once that account is signed in, open the Xbox app again and check if it now stays with the correct one.

If it still immediately restores the unwanted account, another reported approach is clearing the locally cached Xbox Identity Provider credentials. Repeat the step above up to before restarting your PC, instead do this next:

1. Open Settings > Apps > Installed apps > find any Xbox entries
2. Click three-dot > Modify > scroll down and click Terminate > Reset
3. Do the same to Settings > System > System components > (Xbox) Game Bar
4. Press Win + R > enter userprofile%\AppData\Local\Packages\Microsoft.XboxIdentityProvider_8wekyb3d8bbwe\AC\TokenBroker
5. If you see Accounts and Cache folders > delete those folders > restart PC and continue with steps above.

So if you have Xbox Game Bar enabled (pressing Win + G), I'd try that method first, since it specifically resolved a similar case. If you don't use it or it's not signed in, skip that and do the rest of the suggestions.

If the issue persists, let me know in the replies. This first answer might not resolve the issue, but I'll try my best to help you with this.