You did not lose your free domain opportunity.
Are you using a free trial of shared hosting? If so, you will have to pay domain registration upfront because those are not trial, when you register a domain name you have use of it for one year. When you switch from Free trial to paid plan and actually make the payment you will get the option to make that your free domain, and what you already paid will come off your total due at that time. (If you pay but do NOT continue on to paid hosting after the free trial for whatever reason, you can set that domain up for usage at another host, the domains you register via dreamhost are not tied to dreamhost, they can be configured otherwise.)
If you have already paid for hosting, is your account “pending”? If so, you likely can’t register the free domain until it’s out of pending.
The Manage Accounts page in the dreamhost panel has a section titled “Domains Registered”. The words right underneath that title will tell you if you have a free domain available, use the button right below that to register your domain.
If you feel the information provided there is incorrect, open a ticket or chat with support.