You do need to do a complete site redirect to the new one.
I'd advise to keep page URLs the same (except for the domain) to make it really easy with a few lines of .htaccess code.
Also, deleting content won't help it. On the contrary, it might help G notice that the new site is the copy.
Here's a process to consider:
1. Get new site set up to look identical to old site
2. Get new incoming links similar to old site: try to change the strongest existing links to point to your new site (it's identical to the old one, right?) and get good links to your new site
3. Do a 301 permanent re-direct from old site to new
4. Relinquish old URL if I have to (timing concerns?): try to have 301 redirects for as long as you can.
5. Verify the new site in the Google Webmaster panel and submit the sitemap (
read this help from G)