Unless things have changed big-time since we went in 1999, you will sit with other people. They'd rather serve full tables and leave some tables empty, then put people at every table so they're partially full. One recommendation -- if you want to avoid children, go to the later seating. If you have children, I recommend the earlier seating as everyone with children goes then and it's noisier/more comfortable for families.
We were on our honeymoon in 99 and they originally sat us with a family with three children at the earlier seating even though I'd request the later -- not very romantic. The cruise knew in advance that it was our honeymoon, as there was a congratulations basket and a note form the captain in our room when we got there. We requested a change to the later seating and they sat us with a couple where the man was doing a lecture tour on the cruise -- they didn't like Disney and weren't happy with the ship at all. We were especially bummed because there were empty tables around. We planned Palo one night, which was wonderful because we sat alone -- that's one way to have a non-drinking table for at least one meal. I'd have paid the extra $10 every night if I'd known that ahead of time!
But the food was outstanding and we didn't see anyone drinking to excess, so I hope you'll enjoy youself no matter what!