Using classic Outlook for Windows in educational settings
I was recently forced to switch to the "new" Outlook and have found the same problem. The "Templates" feature is basically Quick Parts without all the features allowing you to share them across Outlook and Word.
I found a workaround for my custom Outlook templates that is allowing me to create a new email using my templates. I opened my custom template folder in File Manager and saved them as a shortcut to my desktop. It's not as convenient as my Custom Form button in Outlook, but it's faster than going to the folder each time and it allows my attachments and images to remain in the email template.
Find your templates in c:\users\username\appdata\roaming\microsoft\templates (change "username" for your specific username), then right click on the .oft file and Create Shortcut. Move the short cut to your desktop and double click to open the template. It should open the email template in the new Outlook and allow you to edit and send it.
Hope this helps!