Hi Jamie -- At this time, the Story Map Builders don't support sharing a story directly to a group. But you can share stories to groups manually from your Content, as you expected. The thing to note is that story maps contain maps and layers, and all of the story's maps and layers must be shared to the group as well.
When you share a story with your organization or with everyone from the builder the builder takes care of setting the sharing permissions on all of the maps and layers. But when you want to share a story to a group you need to share the story as well as all the maps and layers to the group from your Content page. If you don't own some of the maps and layers in your story you'll need to invite the owner(s) to the group and ask them share those items to the group.
Here's an article with some additional information on working with story maps that might be helpful to you.
Owen Evans
Lead Product Engineer | StoryMaps