We camped there in October. Nights pretty cold at that time but otherwise it was great. There are 4 locations to choose from; Carey's Reserve which is on Burrinjuck Dam foreshore when the water is high (over 90%). Unfortunately now it is more like 40% so no water , and not much other reason to camp there.
The main area is called Billy Grace Reserve. There is a ranger onsite where you can pay fees and I think they sell ice too. BGR is by the Goodradigbee River and has some excellent camping but is also the most busy/popular. I think there are some showers here but I didn't see them.
Up the road 5-10 mins is Swinging Bridge, also on the Goodradigbee. Has nice river access but is small and all the sites are in a narrow strip between the river and the road. It would get quite dusty with passing traffic and small children would be a worry.
The best IMO is Micalong Ck which is 5mins further along. This has excellent open camping on well kept grounds adjacent to the creek. There are toilet facilities but no showers. It is the most picturesque of all four locations. The only downside is that Wee Jasper Reserves are all popular, well known and easily accessible, so it will be busy on any holiday weekend. If you can get there during the week away from school holidays you might actually get some decent space to yourself. They have a Facebook page.. check it out.