Have you been to Disney World ever before or would this be a first experience?
Animal Kingdom is an easy park to get character interaction. They regularly have Mickey, Minnie, Goofy, Pluto and the Pooh gang out and about (plus others). They have some great shows to experience but I'm not sure how well that would be appreciated by a 1 year old. This park always feels "hot" to me as there are not enough shaded areas and the walk from one area of the park to another can be quite lengthy.
EPCOT is nice in that every country in World showcase has a station where kids can decorate masks or learn a word or two from the countries. But again, a lot of walking. The future world part of EPCOT has a lot of hands on activities that your boys might enjoy.
Another suggestion would be to go to one of the water parks for the day. The boys could enjoy splashing around in the water without dealing with as many crowds and long lines.
Finally, if you are just looking for something to "kill time" another option would be to go to the Hyatt at the airport and rent a day room. Then you would have the relaxation of knowing you are already at the airport. You could take a nap, shower, change, enjoy the pool, etc which I think most young boys love and have a very relaxing end to your vacation.
Good luck with your choice.