Be sure to check if RVs are allowed on that section of LSD (Lake Shore Drive). Parts of it are closed to RVs - even to pickup trucks!
Since you have the family with you, another couple of ideas: Indiana Dunes State Park in Chesterton, IN or Illinois Beach State Park in Zion, IL. Both are about 45-50 miles out of the city but are along commuter rail lines that will take you right into downtown. The parks have electric hookups (also some water hookups at IL Beach) and both parks have decent beaches in them (Lake Michigan). Although IMO Indiana Dunes is a bit nicer CG, both are decent places to stay. Bring bug spray, though. That area of the country has LOTS of mosquitoes.