That's the beautiful feature of root relative urls, you don't have to specify a domain name. With my plugin activated all urls will look like this:
Which means when you can access your site in any of the following ways:
(fake ip used as an example)
Browsers, search engines and most everything that handles internet traffic will know to re-use the domain name you used in the browser address field when it encounters a url that starts with a forward slash "/"
And if you ever need to migrate your site to another host, or perhaps you need to add a domain with a country specific extension to your site, you can do so without changing your content.
Please note, that if you have existing content the links will still have the domain name in them. Unfortunately I don't have a fix for that just yet. My best recommendation is to either do a search and replace on "http://www.olddomain.com/" and replace it with "/" (be sure to include the "/" in the search term.) as described here: http://codex.wordpress.org/Changing_The_Site_URL#Changing_the_URL_directly_in_the_database
Another option, if you are not comfortable with the above, would be to edit the content by hand using the html editor. Just remove the "http://www.olddomain.com" portion of the urls and images/video links. That can be a lot of work of course, but this is also something that experienced WordPress contract developers can knock out in under an hour, so maybe paying someone $50 dollars or so to do it for you is a better option if our site is really big.
She's only days away at this point, but I'll do what I can to help you through this :D