Hello tarr001,
We definitely want to ensure you're able to easily stay in contact with others using your Mail app. Let's take a look at this together.
From your post, it sounds like the Hotmail or Outlook account is still logged in on your device, since you were unable to log out. In that case, let's go to Settings > Mail > Accounts > Outlook/Hotmail account > Mail. Is that enabled?
When viewing your Mailboxes list, do other accounts appear there, or is it empty? To view that, check out: Use mailboxes to organize email on your iPhone or iPad - Apple Support
Additionally, when attempting to log out or in to your account, do you receive an error message? If so, what does it say exactly? When you go to Settings > General > About, what do you see for your software version? This can help us target our troubleshooting and get this resolved for you.
Thanks!