There are two provincial parks on the gulf side I can recommend. The first is Val Comeau http://www.tourismnewbrunswick.ca/en-CA/Pr...8118044604b.htm
where you can see the fishing boats come and go from your campsite, as the sites are towards the river, and to get to the beach you only have to cross the dunes.
The second one is http://www.tourismnewbrunswick.ca/en-CA/Pr...8C0AD107CAB.htm
It is situated on a cliff overlooking Northumberland Strait, and on clear days you can see PEI and the bridge.
Both parks have a beautiful beach, but sometimes you can encounter jellyfish. There are only electric hookups, but a dumping station and water points are provided. They are not very far from some smaller towns, but to my knowledge there are no kids programs. Both parks have only a few shade trees, but because of the huge space between sites one has sufficient privacy. As for the shade, with the constant breeze from the gulf, especially at Murry Beach, you won't need it.
One park where the beach is not quite as nice but you are closer to the action of Caraquet without being disturbed by traffic noise is http://www.campingcaraquet.com/english.html
For mor info, try http://www.tourismnewbrunswick.ca/en-CA/