You might want to provide a link to download that problem widget/plugin and maybe someone can see the problem.
Nothing really to see. If I drag over my new widget, everything in my sidebar is gone.
Actually wanted to know WHICH widget and if the widget is provided by a plugin, what plugin.
Deactivate all plugins, change to the WordPress Default theme. Can you drag and drop widgets?
It sounds to me like what you are calling ‘widgets’ is actually a hard-coded fallback sidebar that shows up when there are no dynamic widgets in use. When you go to the widgets management section, do you see any widgets in the sidebar besides the one the added?