You aren’t out of luck. It is just going to take some time. I’ve always been able to get what we needed eventually, even for large parties. Here are my suggestions.
Check out the Touring Plans reservation finder. It is free. You sign up for the meals you are looking for, and they send you an alert if they come available. Some people really love it. I’m not a big fan as I never make it to the computer fast enough when I get the alert, but it doesn’t hurt to try it.
What I do instead is the refresh method. You basically pick an ADR, and then try repeatedly to get it until you do. Open a page and leave it in the background. Periodically throughout the day refresh it. Sometimes it takes a few days, but you can usually pick up someone’s cancellation eventually. This strategy works well if you have an office job because it’s easy to pull the screen up and refresh during breaks, or while another program is loading, or you are waiting on something else, just keep hitting refresh.
The other thing to keep in mind is not to get discouraged. People are bad about hoarding ADR. They will make multiple ADR 180 days out that they do not plan to use. Then when they book their fast pass at 60 or 30 days, they will drop the ones that don’t fit in their schedule. Some people hold extra ADR right up until before the no show fee will go into effect to make sure their plans aren’t going to change, so sometimes the best time to book ADR is last minute.
You can also try splitting your group up and searching that way, a table for 3 and one for 4.