I'm not on core but I am doing South Beach Diet. When I was at WDW a year ago September I was in early phase 2 (I had started the program about 5 weeks before the trip). I did entirely counter service meals and some of the things I had that I think would be core friendly also would be:
Grilled Salmon with vegetables and rice from Tusker House (AK). In September I asked for double vegetables and hold the rice.
A Salad from the Earl of Sandwich. I know there are salads all over WDW but these guys use mixed greens instead of Iceburg.
Vegetarian Chili at Columbia Harbor House.
Chicken Wrap at Pacos Bills.
I hope that get's you started. My suggestion would be:
1. Take what you've learned and make the best choices you can.
2. Don't worry if you are eating slightly off program.
3. Have fun. You'll be doing a lot of walking and should have some activity points to play with.
This was the tactic I took in December. I've been to WDW twice since starting SBD and haven't gained weight either trip (I even lost the first trip)
Hope that helps.
Jeremy