Can I keep the same embed URL but change the image?
No, but...while individual 360s will have unique embed URLs, if you wish to send a single link (e.g. to stakeholders) but update its contents, consider using a Collection.
Collections can contain a single item (or more), but crucially the share link stays the same no matter what the contents are. They're also embeddable and share-able the same way. So for this use case, create a Collection with a single item, and then when you want to update the item, just update the contents of the Collection.
It's also an easy way to share multiple versions if you wish to keep the old versions as well.
If you are looking for a way to avoid re-creating Annotations, Hotspots, etc for a newer version of an image, use the Move/Copy Annotations feature instead. It will enable you to move all your annotations and update navigation hotspots with much less effort.