Hi Jen -- I'm not able to reproduce this issue with either a Web AppBuilder or Operations Dashboard app embedded inside a story.
Just to clarify, any extent changes you make to a map that is part of an embedded app are not saved when you are in the story builder. You need to update and save the configuration of the embedded app to update the map extent. Also, could you expand on what you mean by "leave the page and come back?" Do you mean you switch to another browser tab and then back to the tab with the story, or in the same browser tab you leave the story map to visit another page and then return to the story map? Lastly, does the embedded app behave the same way if it's not embedded in a story?
If you are able to share your story with everyone and provide the link it would be easier if we could take a look at your specific story. If you're not able to share your story I'd recommend contacting Esri Support and they can help you troubleshoot further.
Owen Evans
Lead Product Engineer | StoryMaps