Hotel management doesn't handle dining reservations. The concierge desk can book them for you if any are available but pretty much same basis as your finding what you want yourself.
As prior posters indicate, keep checking or sign up for an ADR alert service.
Now that people can cancel up to 2 hours before their ADR time, I bet there's a much better chance of snagging a cancellation late the night before or early on day of.
All such go to a distant call center. Ridiculously, even those from resort rooms are answered at the call center. That's why resort guests are advised to go to the front desk about any issues.