I wouldn't worry at all, RCCL and Celebrity seem to "run out" of slots quite early, but that doesn't mean there isn't actually space. If you used a travel agent, make sure that he or she has properly annotated your reservation with your desired dining time. Once on board, head to the MDR and ask for a head waiter. It seems like on Royal, in addition to every line I've sailed, they're able to squeeze you in without an issue (assuming you're not sailing with a large group or anything of that nature).
Report back after your sailing, but I'm quite confident this will work out.
