In regards to entering Sandy Hook?
They don't always close the park gates between 11-2 on Saturdays and Sundays, but they have to if they reach capacity. They count cars going in and out and when the number of cars in the park gets too close to the total number of spaces in all parking lots, they have to stop letting cars in until people start leaving. This is more likely to happen on really nice weekends and on holidays.
Best way to deal with this is to plan to get there either by 8 am or after 3 pm. If you go in the morning, get there as early as you can to make sure you get a spot in Lot G closest to Gunnison. Sometimes that lot is full by 10 am regardless of what's going on at the front gate.
If you can't get there that early, going later still gives you plenty of sunshine and even a sunset to enjoy. I think the gates are open from 5 am to 9 pm, but I think you can stay until 10 pm, so if you time your travel right there's plenty of time to get in either before or after the midday crowd.
They close the park any time they reach capacity. Usually there are posts notifying you of this on the roads leading to the park. It gives you warning do you can plan to go elsewhere. I have seen closings during the week but it is rare.
This is the reason why I've been hesitating all these years to go to Gunnison on a weekend. You're taking a risky chance getting up early in the morning to go to make it in time especially for me that I'm two hours away. On the other hand, you have a choice of leaving to go the day before. However, you would either have to drive all the way home after a long tiring day on the beach. Or, pay for 2 nights at a decent hotel and end up spending at least $300 for what it seems to be for most hotels in that area. So maybe on a weekday but is there enough people to make it somewhat enjoyable on a nice day that is?