Hi @DanielBarwick,
Could you possibly post a screenshot of what the announcements area in your current course looks like, along with the dates area of one specific announcement when in edit mode?
What I think you may be seeing is a result of how the announcements may have been done in the original course and how you import them over. If the announcements have no dates or the dates are in the past and you just import those without adjusting dates, the "posted on" does appear to end up blank. That doesn't mean that students can't see the announcements, but they wouldn't get an email/app notification because there is no date or it's in the past. If the announcement date in the original course is in the future though, you should see that come over and be displayed as a "delayed until". In addition, if your announcements have dates in the original course and you adjust the dates as part of the import to make the new announcement be in the future, you should also see that new date as a "delayed until"
Here's are example screenshots from my tests:
Original course:

New course:

The "Hi" announcement had an available from date set in the past. So when I imported it to the new course, there is no date shown on the main announcement screen because the "post" was in the past so never really happened in the new course.
The "The future" announcement has an original "available from" date in the future, and that imported and shows the same future date in the new course.
The "The past to the future" announcement had an "available from" date set as May 1 in the original post. it shows as posted July 1 because I just created it today for testing purposes. When I imported that announcement into the new course, I adjusted the dates and chose to adjust May 1 to Aug 7. That caused that announcement to get the "delayed until" applied on the main announcement screen.
If you want announcements to be posted on a schedule for your new course, the key seems to be to ensure the announcements in the original course have an "available from" date set on them, and to use the adjust dates feature during the import to ensure all of the adjusted announcement dates will be in the future.
I hope this makes a bit of sense and helps you a bit. If you don't have "available from" dates set, that may be a big task to go add, but it would definitely help out if you plan to reuse those announcements for multiple future courses.
-Chris