I appreciate any help so that I can get them in the coop before they start laying eggs.
I don't know how much trouble you are in with this. Hens don't always lay where they sleep. Plenty of us have had hens hide nests outside, Growing up, Dad had chickens that slept in trees but laid in the henhouse. Many of my pullets seem to learn from the hens where a safe place to lay is. I sure can't give you any guarantees that they will lay in the nests with the hens even if they are sleeping in that coop, especially with them free ranging with all the great places they can find to make a nest, but even with them sleeping elsewhere I think there is a reasonable change they will lay in those nests. My preference would be for them to be sleeping in that coop when they decide where they will lay but I'm not sure it is an absolute requirement.
When they are free ranging you say the pullets can't get any closer to the other hens than three feet. You are kind of in the same situation inside the coop when they are going to bed until they mature enough to sleep with the adults. They need a safe place to sleep where they are not going to beat up by the others or they will not want to sleep in there. You don't want that to be the nests, you won't like poopy eggs. That's the main reason you need extra room when you are integrating, they need some separation.
I'm not a believer in magic numbers for square feet per chicken or many others, if you follow the link in my signature you may see why. I've seen recommendations from 1 square feet per chicken all the way up to 15 square feet per chicken on this forum, who do you believe? I understand that people need guidelines when they are starting out and 4 square feet per bird is a reasonable starting place. For some people it is overkill, for some people it is not enough. Your situation is an example. With five chickens 4 square feet per chicken doesn't give you a lot of separation when integrating. If you had 16 chickens it would give you room where you could separate.
So what can you do? You probably don't have much time before the pullets start laying and building material are really expensive right now. I think them free ranging together during the day helps a lot, even if they stay separated.
In South Carolina they don't need a lot of weather protection, especially in summer. The roosts need to be higher than the nests which might complicate it, the far end of that shelter doesn't look very high. You should be able to modify it by stripping out the floor and front and maybe modifying the roosts. Leave the nests. Adding a roost at the far end from the nests might be enough. For summer especially just cover the top parts of those wire walls with something to block the wind and blowing rain during storms. A tarp could work. It does not have to go all the way to the ground. That could give you time to do something more substantial for winter.
If roost height or separation is a problem you could remove the nests as well and either lower them or even build new nests on the outside and cut a hole in the wire so they can get to them from the coop. Spread your roosts out more. I use the tops of my nests as a droppings board and scrape the poop off of them. Some people on here don't like that idea. In all this think about your access to the inside, that might get awkward.
If I were building a new coop from scratch and buying new materials I'd make it 4' x 8' for five birds. The more inexpensive building materials come in 4' or 8' dimension. If you plan your coop around that you can avoid a lot of cutting and waste. If you already have materials use those dimensions. It is probably more than the absolute minimum you need but people don't complain about having too much room but they often complain about not having enough. Some extra room gives you flexibility when something comes up, something like integration or if you need a broody buster. Having room can greatly reduce stress.