Hi @Martina Vikefors ,
Welcome to ur forum!
When I switch to my other organization that I am a consultant for
As I know, we could switch to other organization like blow, is it your case?
If it's also your case, according to my tests, I have tried to configure the main account in Outlook and switch to other organization and back like above in Teams, but my account could still work noramlly in Outlook, it seems to I couldn't reproduce your issue.
For the issue about "Need Password", in the point of Outlook, what's the version of your Outlook(File>Office account>About Outlook), please first ensure that you have updated to the latest version of Outlook.
After that, we could try to create a new Outlook profile and readd your account to avoid the interference of corrupted Outlook profile. For more information about how to create a new Outlook profile and work with another profile, please refer to this article : Overview of Outlook e-mail profiles.
If the issue still exists, based on my research, I found a similar thread that mentioned DisableADALatopWAMOverride registry value may help us fix the "Need Password" issue, please kindly follow it to create the registry value to check if it could also be helpful to you(Note: Serious problems can occur if you modify the registry incorrectly. Before making changes, back up the registry to restore it in case something goes wrong.)
If that's invalid, considering corrupted Outlook client may also cause it, it's recommended to working with SaRA tool to troubleshot your Outlook to check.
If your issue has any update, please feel free to post back.
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