Burlingame is huge. We were there the weekend after Labor Day weekend in 2014, and it was pretty quiet. Nice for biking, cute little beach, dog friendly. Bring quarters for the showers. Westerly is nearby with a 24 hour WalMart in case you forget anything (which we did, being pretty new to camping at the time!) also, I got stung by a wasp, so watch out for those guys.
We love this campground, and have been coming here for a few years. It’s dog friendly, and even though dogs aren’t allowed on the swimming beaches, you can bring them to different parts of the pond for swimming, and no one gives you trouble. The restrooms are clean, and the showers have hot water. There’s two large ponds for swimming and boating, and paved and unpacked trails for hiking and biking. The Cape Cod Rail Trail runs through the park, and is a great way to get around. The park is large, with about 7 different camping areas, and it always feels quiet an peaceful, even in the middle of a busy tourist season on the Cape. Oh, and you can have a campfire, you just need to buy your firewood at their adorable camp store. I camped at a different campground on the Cape years ago, and campfires were not allowed, so being able to have one is a big plus for Nickerson!