I have been able to reproduce this problem only once on a demo site with a lot of plugins and some sketchy themes installed. When testing this plugin by itself or with a few other tried-and-true plugins, it works flawlessly.
So I think it may be a conflict with another plugin - and that's going to be hard to track down.
The simplest workaround is to deactivate and reactivate this plugin after adding/updating a page and before editing a widget. (@escalatoraccidnt, let me know if that still doesn't work for you.)
Why does that work?
Display Widgets uses a form of caching called a "transient" to store a list of the current pages. It's a smart way to go. Whenever a page is added or updated, the transient is deleted. Then a fresh transient with a new page list is created the first time you edit a widget.
Something seems to be preventing the plugin from deleting the outdated transient. De/activation creates a fresh transient.
An admin option to manually create a fresh transient, again after adding/updating a page and before editing a widget. Not ideal, but simpler than de/activation once we get the code installed.
Let me know if you are interested in Workaround #2.
You are not alone in this - see this thread - but it's impossible to reproduce given the number of plugin and theme combinations.
In order to track down the conflict, we need to verify that the problem is indeed the inability to delete the transient and what's stopping it.
If you would like to participate in that bug hunt as a way to help others experiencing the same problem and to give back to the author, let me know. I can build a simple plugin that records the key events to a log file. It would also list your installed plugins. (No other site or personal information, just lines like "transient contains 7 pages" or "transient not found".)
You would then install the plugin, perform a page update then edit a widget, and email me the log.
I realize that's a lot to ask and may feel a bit intrusive but the problem won't be fixed without more information.