This is an ongoing problem with modules and is made worse when developers fail to appreciate the finer permissions granularity needed in a business or any multi-user system, and simply think that everyone should be an admin. You have to assess every module for the ability to manage content without having page edit or admin rights. Most modules fail the test entirely or only allow a subsection of their CMS functions to be accessed.

However - From what I recall, the core announcements module does let you do everything with only module edit rights. I had a couple of people using it a while back and I never provide admin or page edit rights to content creators. Can you check that your container provides proper menus for this. It needs either the action menu visible or preferably also action buttons to be enabled in the code. One got'ya feature: If you disable a module container, all of the menus will vanish for edit users. Therefore it is best to create a special blank container that still contains menu items and use that instead of nothing.

Regards
Rob