James,
That's not supposed to happen. When you preview the map and tap the "SAVE" button, the app is supposed to take you to the map list where your just created map should be visible in the list. Then you can tap the newly created map in the list and see your location on it if you are in the area of the map you just created.
What version of Custom Maps do you have? Some very old versions stored the maps in the "public" area of the phone storage, but Google has blocked that option in recent Android versions, so the latest version from Google Play Store stores the maps to the apps internal private storage area. The maps can be easily exported from there either all at once using a menu option in the list, or by "long-pressing" a map entry in the list and selecting "share map" from the popup context menu.