I've never been there in June, but in July and August, you can't do 2 parks in a day and see everything, even if you're there from morning until closing at midnight. If you have to skip one of the 4 Disney parks, skip Animal Kingdom. Also, since you will be in the Bahamas the second week, you can skip the water parks and just concentrate on the theme parks. Besides Disney, you should go to Universal and Islands of Adventure. Again, that may take you 2 separate days. Also, Sea World is worth seeing if you've never been to one before.
As for eating at Disney, after numerous trips, we found that we enjoyed relaxing for meals and taking a break to eat at the Disney hotels instead of eating fast food at the park. It's more expensive this way, but you get to see the other parts of Disney and the transportation is free. Take a boat to Grand Floridian for lunch, or Polynesian Resort or the Wilderness Lodge from the Magic Kingdom. If you're at Epcot, just eat some ethnic food at one of the places in World Showcase. At MGM, there are many themed restaurants, but we really enjoyed 50s Prime Time Cafe. That's usually crowded, so you should make a reservation as soon as you get to the park or before.