WordPress.org

I can see icons, but the text of all my links has disappeared!

You are probably still using wordpress' default Links widget, or a call to the default wp_list_bookmarks(). To see both text and links, you will have to use the new list_iconized_bookmarks() function, or the Iconized Links widget.

Do I have to use N.I.B. to display the bookmarks with icons?

No, you can use your own plugin. Though if you use the default wp_list_bookmarks, you will not be able to show both favicons and the link texts.

If you want to display the bookmarks with favicons, you can use either the new function list_iconized_bookmarks or the Iconized Links widget that should have appeared in your widget list.

I have uninstalled N.I.B., but the icons are still showing in my sidebar

If you are using the standard Links widget, images will be displayed if their address is available. If you have used N.I.B. to find favicon addresses, those will still be saved in your Links' Image Address field. N.I.B. does not clear your images when it's uninstalled, because that would be too severe a change in your database. You can either disable the show_images option for wp_list_bookmarks or manually remove the favicons from the Image Address fields.