Using New Outlook on Windows for personal email, calendar, and contact management
I finally found the new solution that works for both the desktop versions of New Outlook and the web version. There is a "waffle" style button in the Message ribbon. When you hover your mouse over it's labeled as "Enhance Outlook with Apps". Click on that button and one of the apps in there is "My Templates". I'm not sure why Microsoft decided to add an extra click to the workflow but at least the templates are available again.