Hi Jeff,
Just to clarify, please answer the following
- Are you also accessing your Outlook emails using the Outlook website (http://outlook.com)?
- If you're accessing your Outlook email using the Outlook website, what web browser are you using? (Examples of web browser: Internet Explorer, Microsoft Edge, Mozilla Firefox, Google Chrome, etc.)
If you're using any of the web browsers, you will be to see the "Keep me signed in" check box when you are logging in to the website. After you enter your Microsoft account, it will ask you for a password. On that page you will see the "Keep me signed in"
check box. Please refer to the screenshot below:

If you have the same concern as the original poster, you can resolve this by clearing your browser data or by resetting your browser's settings. If you're using Microsoft Edge, you can clear your browsing data by following the steps found on this
link. If you're using Internet Explorer, it's best to reset your Internet Explorer's settings. To do this, follow the steps found in this
article.
If you still have any questions, kindly reply to this thread.