I managed to fix it. Thanks to jnjroach for getting me on the right track. Will list how I did so for future reference to others who may have exhibited this problem.
First, create a System Restore point, just to be safe.
Open up File Explorer, right-click in the navigation pane and select "Show Libraries". Right-click on PICTURES library and select PROPERTIES.
It'll show a window with your picture library folder locations; if you have a few listed there, or perhaps none at all, that's what's likely causing the problem with Modern IE. Click on RESTORE DEFAULTS, and then APPLY. It should only show one folder there now, your default Pictures folder. Go back to Modern IE and test to see if pictures save.