We took our 1.5 year old son on the western cruise in January, and are doing the same itinerary in January of 06. If you want to try for early dinner, first thing, when you get on board, go right to the area that they are taking Palo reservations and see the head server from the dining room. We did this and we were switched to early seating. We used Flounders and it was great! A couple of times through the cruise, they'll have times that they will be open for you to take your little ones in to play. You'll have to stay and supervise, but it's a nice time for the kids to get familiar with the surroundings and you can meet and visit with other adults. You can also take children under 3 into Oceaneer Club to play and participate in their activities, but you have to stay with them. This was saving grace for us! We never minded staying with Trevor and enjoyed our time playing with him. If you don't want to take strollers, Disney will have small, umbrella-like strollers available on a first come, first serve basis. They were really nice strollers! Diapers were very expensive on board - make sure you have plenty. Children who are not potty trained will be able to play in the splash pool located in one of Mickey's ears. When we sailed, staff were always keeping a very close eye on children who had swim diapers on and they would "shoo" them out of the other pool areas if they saw them. We had a hard time finding snacks on board for our son. Either plan to grab an extra cereal box at breakfast or take your own snacks along (we bought craisins and Goldfish crackers in Key West). This isn't to say that there is a lack of food, but we didn't always want Trevor to have ice cream or french fries when he was hungry. You'll LOVE the free drink station on deck 9. Each night before we went to bed, we'd take the sippie cups up there and fill 2 with milk to put in the cold box in the room overnight. That way, we had drinks for Trevor first thing in the morning.
Hope this helps! You can PM me if you have other questions.
Chris