Which option to move WordPress

I have finalized my new WP site on Westhost’s servers, at URL http://[IP address]/~[username]/. I would need to move the WP site to http://www.mywebsite.com so that the links work on the new domain. I have my old non-WP website (with GoDaddy) on http://www.mywebsite.com which I am retiring, replacing it with the new WP site.

I understand that I need to point the DNS to Westhost’s servers, which I should be ok with. But it is unclear to me which option of moving WP I should choose, and at which point I should do the DNS change – before or after. I’ve gone through the Codex and this step-by-step guide, but I’m still unclear on which option is the right one for me?