In addition to unlocking, you need to make sure it’s not an anonymous registration, such as GoDaddy’s Privacy Manager, or DreamHost’s Anonymous WHOIS service.
You also need the Authorization code from your old registrar to prove it’s a legitimate request. It’s usually a 10 or so digit code.
You don’t need to update DNS, but the sooner your DNS says ns1.dreamhost.com, etc., the sooner you can host your site here. A registration transfer may take a week, but making that DNS update only takes a day or so.