Maybe sign up for a basic access/free Touring Plans account and use their reservation finder? I just used it for the first time and it worked really well for me.
I wanted to switch our Be Our Guest breakfast to lunch, but nothing was available when I checked. I saved a reservation search yesterday and got a text this morning with a lunch reservation time. I booked that, then set a search for an earlier time just in case -- within an hour I got another text and was able to replace my reservation with an earlier one. I would expect it to take longer with a party of 7, but it's worth a shot, and it's free.