Update: I'm halfway through with a solution. In the two php files, tadv_admin.php and tinymce-advanced.php, the plugin is using the following references:
WP_PLUGIN_URL
WP_PLUGIN_DIR
Those are where the plugin is getting the original subdirectory instead of the new domain. So, to replace it, I did the following.
Find all instances of the WP_PLUGIN call, such as this on line 63 of tinymce-advanced.php
$wp .= ',' . WP_PLUGIN_URL . '/tinymce-advanced/css/tadv-mce.css';
and replace with:
$wp .= ',' . get_bloginfo('wpurl') . '/wp-content/plugins/tinymce-advanced/css/tadv-mce.css';
Note, I added the get_bloginfo('wpurl') and also "/wp-content/plugins" in front of the /tinymce-advanced
Now in wordpress, the popups seem to load. However, the information in the popup isn't being replaced, so all the labels are in {brackets}. Again, this is only on the sites with domains attached. I'm still working on a fix for this.