Hi there!! We sailed with my DD when she was just barely 1. She is 6 now and has been on 5 cruises and loves them! At the 1yo stage, she stayed with us most of the time as the drop off kids club only starts once they are potty trained. However, they do have a play area for parents to stay with the little ones and play with a bunch of toys. Also, the splash pad area was really important as kids with diapers are not allowed in the pool or slides area. So yes, as mentioned above, it was work, but it was still fun.
At the 3yo stage, it was a totally different experience. She went to the kids club and loved it so much she wanted to stay all the time! We really only saw her for meals and tours but she had a BLAST! Royal's kids clubs are top notch (As I'm sure Disney's are too!!! ), also the Liberty has the Dreamworks characters so we had some really good character interaction if we wanted.
On the Liberty, I would suggest getting a Promenade View Interior. That way, your kids can watch all of the parades without having to deal with people standing in front of them or smushing them. I have stayed in the PR interiors several times, and I always choose them if I am sailing a ship that has them.
I hope all of this helps, I have cruised Royal quite a bit, and at least four times on the sister ship to Liberty so I am fairly familiar with the ships if you have any questions.