How Does Shared Location work?

The share locator asks whether the visitor wants to share their location.
If they say yes, the database will check the closest location to them and serve that page.
Once they told their browser they are happy to be taken to their closest location, the browser will remember that and honour that request every time (unless cache etc is cleared).
If there are no listings in that location, it cannot show any. Best practice is to add locations as you add listings, that way you do not have empty pages.

If then changing using the location switcher, they should go to that page.
What you are describing is not normal; simply if it works different as on http://wpgeo.directory/ then there is something else wrong.

Post your URL if you like, then we can have a look.

Here is how you clear your shared location:
Firefox: https://www.mozilla.org/en-US/firefox/geolocation/ [6th question]
Chrome: https://support.google.com/chrome/answer/142065?hl=en
IE: http://windows.microsoft.com/en-au/internet-explorer/manage-delete-browsing-history-internet-explorer#ie=ie-11

You can also disable the share location plugin.
From my testing your site works as expected regarding locations, Chicago must be my closest location.
The problem is I think that your map and/or pages are being cached, so it looks like you are not taken anywhere else because the map does not change.
And depending where you navigated to, there is some redirecting going on.

Disable your plugins one by one, start with the ones that might be caching like wordfence.
Delete your saved location settings too.

Let us know if you have any luck, or we will have a closer look.

I would start following Guust’s suggestion than if any of those plugins isn’t the culprit ask to your hosting provider for support.

I tested it too and it works as expected for me too, except the map being cached.

Thanks