We cruised on the Paradise when our boys were 1 1/2 and 3 2years ago. We had a great time! We went a bit off season, so there were less kids on board, but we still had a nice time. It doesn't matter how "geared" a cruiseline is toward children if the children can't be in the program!!!! Carnival includes 2yos in the program and changes diapers. Children younger than 2 can also be in the group babysitting and enjoy the play equipment.
We were on the Explorer on April 20 cruise this year. Our boys are both now over 3 and fully potty trained, so fully able to participate. There were MANY children on board that were too young or not trained so unable to participate. One mother that I spoke to had a 4 yo and 1 1/2 yo. When she dropped the 4yo off and the younger one saw the ball pit, she said he cried for about an hour not being able to play in it. He also kept saying "ball ball" the rest of the night.
Carnival will welcome both your children with open arms. The staff I've found to be particularly child friendly (although never had a problem on any other line either). Paradise is not a party ship by ANY stretch of the imagination. For our cruise, the boarding area looked like the lobby of the sun city seniors center since most of the passengers were older. The ship was pretty quiet after 10. Again, we went off season, so less families than normal given the school calendar were on board. Its a nice ship to be on, I wouldn't hesitate to go on it again, and its terrific not gagging from the smoke in the cassino or other public areas.
Enjoy your cruise!