When I create a Story Map Tour using an Excel spreadsheet as the starting point -- from Excel create a map using Maps for Office, then share the Feature Layer and web map to AGOL, then in AGOL create a Story Map Tour from the web map -- the images don't display even though there are columns in the spreadsheet with proper URLs. Apparently Maps for Office renames the columns/fields in the spreadsheet when it creates the map. These renamed fields are not recognized by the Story Map Tour -- it prompts you to point to the fields for 'Name' and 'Caption' for the photos and the field for 'Color' for the icons.
The stickiness/persistence issue could be resolved if you were also prompted for the fields that contain the 'URL to Photo' and 'URL to thumbnail.' These fields should also appear in 'Settings' in the 'Data' tab, which allows you to set/reset these pointers.
[this is an alternative to the solution I proposed in 'Excel, Maps for Office, and Story Map Tour']