When it comes to migration, the ideal scenario would be that you migrate the data from one Droplet to another and then change your DNS.
You don't want to setup two A entries for the same domain that point to two different IP's -- you want to swap the current IP with the new and allow the other to resolve.
You can, of course, shutdown the old Droplet, though that doesn't force the DNS to resolve faster. It will, however, eliminate the potential for visitors that come to your website from potentially hitting the old server.
In most cases, once your DNS is setup (i.e. initial setup of name servers for your domain), changes such as new IP's and new entries resolve rather quickly.