Are leashed dogs allowed on the beach itself? For example, I want to swim and sit in the sand while my boyfriend sits in the sand and shade with the dog.

Dogs are allowed in general vicinity but not on cement walkways , common areas, concessions or beach area. Yes in your campground, remote.areas and hiking, etc. And always must be on leash.

Dogs are only allowed in the camping area

Only service dogs, they're allowed on the grass though!