First of all, the dining package does include Character Meals. You'll have to use a table service meal for those. If you want one of the more upscale restaurants (California Grill, Artist Point, Coral Reef), you'll have to use two table service meals per person.
If you have the cash, Victoria and Albert's as people have said, is a nice (albeit pricey) place to eat. Very nice atmosphere, very elegant.
My wife and I ate at Coral Reef (using two table service meals) and absolutely loved it. The view is breathtaking, especially if you get a seat on the bottom level next to the aquarium. The food is mostly seafood, but it's great. Very nice atmosphere, romantic as you could possibly get in a theme park.
We also really enjoyed Crystal Palace (a character meal using only one table service meal). We told them at the door we were celebrating our honeymoon and they treated us like royalty. When we were seated, our table had about a million little Mickey Mouse ears arranged in the shape of a heart. My wife didn't want to eat because it would mess up the arrangement. Now, it's not super romantic because it's a character meal, but it's one of the more laid back character meals at Disney, so it's easy to enjoy tiny intimate moments.
Now, my biggest Honeymoon tip I can possibly give. Your first day there, find bride and groom mouse ears, and WEAR THEM. You will be treated like kings and queens where ever you go. We got our own boat on the El Rio del Tiempo boat ride in Mexico as well as our own boat on Maelstrom. The Mexico ride is much more romantic (grrr, I hate trolls!!!).
I can't stress enough how much those bride and groom mouse ears did for us. We were given so many perks and everyone in Disney (from CMs to everyday people) offered their congrats. I got so tired of saying "Thank you" that I just nodded and smiled after a while. LOL
Our biggest regret is waiting until three days into our trip to buy the ears.
Anyway, enjoy your honeymoon and remember to leave all your worries behind you.