Hey @jonbarrettwordpressorg,
To answer your question, there are a few things that you can do.
We have built out a small tutorial on how to enable date/time picker son your announcements.
https://www.wp-timelineexpress.com/documentation/enable-announcement-dates-times/
This allows you to enable a time picker alongside the calendar, to allow for more granular control over what order things display.
If you wanted to order them manually, I believe that you could but it would require a small bit of coding. You would want to install a plugin such as https://www.ads-software.com/plugins/simple-page-ordering/.
Once installed you would add a small snippet to your themes functions.php file, something like:
add_post_type_support( 'te_announcements', 'page-attributes' );
Which is right out of the simple page ordering FAQ.
From there you should be able to drag and drop the announcements in whatever order you want. That should display them in a custom order that you set via drag and drop.
