There was an issue discovered in TeleScope version 9.4 because of which when the Admin would try to post an announcement, Telescope would show all announcement lists in the Send to List section, including the lists the Admin isn't a moderator of. But when displaying the announcements in Existing Announcements, it would correctly check if the Admin is a moderator of the group to which this announcement is sent and not display it if the check failed.
This was fixed in TeleScope Standard 2020.4. It is recommended to upgrade to this or a later version. If your TeleScope installation has customizations or you're an OnDemand customer, please contact your Account Manager or support.
After upgrading, try to post an announcement as an Administrator through TSAdmin and you should only see announcement lists in "Send To List" for which the logged-in user is a moderator.