Ok... I'm crossing my fingers, but I think I figured it out.
I have several blogs. This plugin only works on one of them. The only difference between the various blogs is that they all use the "www" prefix in their URL except for one...the one that runs this plugin.
In inspecting the code, I discovered that the plugin is using the non-www URL for the plugin. So... I went and edited this file:
and found the following lines of code:
<h2>' . __('Tweet old post by - ', 'TweetOldPost') . ' <a href="http://www.ajaymatharu.com">Ajay Matharu</a></h2>
<form id="top_opt" name="top_TweetOldPost" action="' . top_currentPageURL() . '" method="post">
<input type="hidden" name="top_opt_action" value="top_opt_update_settings" />
and then changed this part:
action="' . top_currentPageURL() . '"
You would, of course, put your actual URL where it says and of course, if you have wordpress installed in a directory like /blog/ or /wordpress/ you would have to add that to the URL.
It now lets me save changes without dumping me out. Now I will monitor it to see if it actually sends proper tweets.
I am sure there is probably a way to use a WordPress function to call the proper URL rather than pasting it in this way, but I will try to this for now.