Same problem, so frustrating! Long story short, I uninstalled the new version and set the old one up again.
I was also getting an "Access Denied" message if I left the app open overnight. Clicking on another folder, then back to the Inbox cleared that message.
I want to mention that I use a Gmail account so I had that open in the browser to verify if Outlook (new) was pushing to Gmail properly. At least I could tell that drafts and sending was working properly. I also launched Outlook through the browser. Interesting thing I noticed is that, at one point, the PC app and MS cloud were synched but Gmail was not.
I tried many combinations of adjusting the mail handling settings within the app (as per the multiple replies in this post) with no success.
Moved on to Windows Settings > Apps > Installed apps
I tried "terminate", "repair", "reset" and didn't help. I uninstalled and reinstalled the new outlook, seemed to work for a bit but I still had one email that would not mark as read, no matter what I did. I went through "repair" and "reset" under settings, as well as "terminate" to ensure it wasn't still running. No success.
I uninstalled Outlook (new) and saw under installed apps that the old version, "Mail and Calendar" was still listed. I used Search to locate the Mail and Calendar app and executed that and set it up. It's finally marking emails as read properly but only defers the problem until Microsoft forces everyone to use the new Outlook. I agree with other posts that this is absolutely an embarrassment for MS. Fingers crossed they fix this issue before that day happens.
Maybe if MS made a Gmail app for the desktop, our issues would be solved!