You'll like the weather in NYC in June. I would definitely recommend the Bronx Zoo (you can drive and park) or the Central Park Zoo (to stay walking around NYC). Your kids are the perfect ages for the Bronx Zoo and it is so big and entertaining. There is a fee to get in and another for special features. The extra $ is worth it for the butterfly house and kids section, but not mandatory. Also, admission is free on Wednesdays, but it may get crowded. Go early to get parking (any day you go).
I'd bring and expect to use a stroller for the kids. The lighter the better, but try to get a standing deck for the back for the older child (not sure what they call it?). It's also nice to have a stroller for stuff.
I LOVE Lake George. If you are interested in doing a night there I would be REALLY tempted to skip the NJ Beach for a few days at Lake George. It is extremely fun and family friendly for children your age. It doesn't have the waves and beaches of the Jersey shore but it is water front, fun for kids, great for swimming and lots of sand and public parks. And a GREAT Six-Flags water & theme park nearby. They have 2 tot/preschool friendly sections for rides and food. Plus big rides for grown-ups. We've taken DD at age 3 & 4 and will go back. Not sure when the water park sides opens, but it very fun for little kids (and included in admission). And I think it is likely that it might be too early for hot beach weather that makes NJ so fun. I stay at the south side of LG. Plus, less driving for you guys. I'd be very tempted by that.
I love Jenkinson's in PP- they used to bring their penguins out to some of my grade school science classes for me.
Storybook land is great for your girls' age group. If you can find some sort of a coupon code, do it. I love the place but hate the cost. We have been to Brigantine, OC, and AC since Sandy and all of them were fine for the most part. There are some people who are still figuring things out with their homes, but the cities are doing well. It will be beneficial for the tourists to come back and pump some money into the local economies at this point. Some homes in Brigantine will need to be torn down and completely reconstructed, from what I know, a lot of this work is already underway. None of the ones I know about are on the beach or would be a rental anyway.
If you have more specific questions about neighborhoods or hotels as you plan let me know and I will give you the scoop, if I know it.
I'm glad I'm not the only one who thinks it's expensive! But it's so incredibly nice I tried not grumbling about it. I think most Tuesdays are discounted in the summer.