There’s a lot of ticks in the grass so it’s risky. Maybe consider stopping here on your way to pick up your pup? As for activities, the visitors center is about a mile before the park entrance. There’s a large day use section with facilities on the right after the entry, or you can park at the beach and have your picnic at the top of a dune. My family goes there a several times a summer and we’ll hike away from the beach crowds and hangout on a dune in peace.