Hey Mike,
Looks like a great trip.
Yeah, I think you can do the whole trip for about $4K or even a little less.
A couple thoughts. First, staying in NYC can border on the ridiculous costwise. Consider staying outside of Manhattan. For me, part of the fun of being in NYC is using the Subway and public transportation. Brooklyn and Queens offer some pretty reasonable alternative to Manhattan and they're loaded with Subway stops.
Skipping Philly may not be entirely necessary. Given your group, I think you'd be more than glad that you took a few hours to explore Independence National Park, and then strolled a few blocks to South St. Trust me. For your group, Philly won't require an overnight stay. Just head south into Delaware and find a campground, then head down to DC the next day. In the DC area, I've stayed at a couple of campgrounds within reasonable distance to the City.
Myrtle Beach is an interesting choice. Personally, I can't stand the place-but I haven't been there in a long time. Very touristy if memory serves. What about Savannah or other Georgia islands? Take a look into them. You might be surprised. But I know you have relatives in Myrtle, so it's probably not terribly negotiable....
I'm not as pessimistic about doing a road trip while camping along the way as a previous poster. I think it's entirely doable, particularly if the weather cooperates. I would just suggest that you have a minimum of two good quality tents that are absolutely weatherproof. Borrow them or find them used on eBay or something like that. The tents should be lightweight and you should be able to put them up or take them down in less than ten minutes in the dark. And you should have really good sleeping pads, or even air mattresses. Air mattresses are more time consuming but they can be really comfortable. And if you need hotels, based on your route, you won't have the slightest problem finding them. They're everywhere, and often pretty inexpensive (but not like $40....closer to $70).
Food should be pretty easy for your group. It's actually pretty simple. A cooler. Supermarkets along the way. Maybe a camp stove, maybe not. Fast food now and again. And decent meals with your relatives. You'll be fine with that plan.
Whatever you decide, just take your time with the planning process and don't get too bogged down with detail. Have fun....