Late April - early May you might find campgrounds still closed for the season depending on how weather goes in the next few months. The higher up, the more likely for closure until June/July. In the northern sierras, the past few years have seen openings delayed until July due to snow still on the ground or fire damage from the prior season.
Forest service campgrounds won't have hookups, but are great about dogs (dogs can go on trails and beaches unless specifically prohibited).
On the eastern side, you'll have more boondocking opportunities with the BLM lands, if you want to try for a couple nights.
With the kids that young, you don't have to worry about them getting bored - they'll be excited about most everything. :)
One trick I've learned is to choose a campground without a boat launch. You end up with fewer large groups of partyers and a quieter campground - even if the lake has lots of boats.
I don't go that far south usually, so most of my recommendations would be in ElDorado, Plumas, and Tahoe forests.