2 and 3 need to come first. That way the mailboxes will be up and waiting when you do Step 1.
The trick is that due to DNS propagation time, mail will still arrive seemingly randomly on one server or the other. If podunk ISP takes 5 days to get the DNS update, it'll continue sending mail to your old server. superfast ISP may get the DNS update in a day and send mail to your new server.
I suggest that you set up a second mail account on your email client that points to the raw machine name here at dreamhost, and update the first mail account to point to the raw machine name at your old host. NewAddress will fetch from MACHINE.mail.dreamhost.com, as displayed in Account Status in the panel. You'll need to do a reverse IP lookup for your old host's mail server to get the raw machine name for OldAddress.