Never been to Pt. Lookout. We were going to, but when we saw the lot, decided it was too small.
Usually go to Cedarville and it's very nice. I don't know how far you are looking to go, we have a lot of soccer & baseball players who don't want miss games, so we try to say nearby. Cedarville works for that, but it would probably be too far for you if you have the same problem. But they have fishing, hiking and an open field if you want to play games.
Been to Navy Rec Center. It's very nice, but not "real" camping. I mean, when you've got a pool, mini-golf and an arcade...seriously? Also it's expensive.
Been to King's Landing during the day. Seems nice enough. Camping is in a field, not in the woods like Cedarville. Not sure what amenities they offer because we were there for an event and didn't explore the park.
Smallwood, in Charles County would also be some distance for you. Fishing, hiking, camping is in the woods. They have a field for games and a couple of historic landmarks that you can get the ranger to show you. Biggest issue is the distance to the bathrooms. It is a quarter mile, instead of the 300ft max. required to get on the list. For this reason, they are probably going to be removed from the list soon....just as soon as Huntr gets around to it.
Nanjamoy Creek Environmental Center is also in Charles County, also some distance from you. I've been there during the day/evening. Seems like a nice place. Didn't see the camping area, although I saw the cabins, which could make it do-able in cooler weather. We went there to see their observatory. They also have a nature center.
Watkins Park in PG County is nice. They have a nature center, a farm, tennis, sports fields, hiking trail, mini-train tour. Our pack had poor turnout because of oh-my-god-its-PG-we're-all-gonna-die syndrome and then it poured rain, but I wouldn't mind going back there.