I have had them in separate coops and runs about a foot apart for the past 2 weeks and want to move them all into the big coop and run but I'm concerned.
Sometimes integration goes so smoothly you wonder what all the worry was and sometimes there are disasters. The more room you have in the coop, the more room you have in the run, and the more time all that space is available the better your chances but you don't get guarantees. It's just that way with living animals, you can never be sure what will happen.
I also don't want to wait too much longer as the younger ones are outgrowing the current coop.
Totally understandable. It sounds like you have at least three coops, at least three runs, and no chickens in the big coop and run. And that they are all outside now without supplemental heat. Knowing how big each coop and each run is and how they are all connected might give us some help in coming up with specific suggestions for your unique situation. Photos could be a big help.
Without that all we can do is be generic. Aart's list is good for any integration. The differences in their ages might make a difference or it might not. You just don't know.
Without knowing your specifics I'd put them all together on a morning when I can be around to observe. Give them as much room as you can for that first time together. If there is going to be violence give them a chance to run away. It's possible they will all mix and mingle into one happy flock. It may take a little while but they could get there. It's possible one or more of the older chicks could start viciously attacking a younger chick, seeking them out to destroy them. You'd need to intervene. What I think is most likely to happen is that the two groups each kind of stay to itself. If the younger approach the older they may get a peck to tell them to leave us alone so the younger quickly learn to keep a distance. In this case that peck is not a seek out to destroy but just a warning. It may take some judgement on your part. If one or more is going out of their way to injure a chick they need to be separated longer.
Do you consider that run predator proof? If it is you may be OK with some of them sleeping in the run for a while. Predator proof is a big issue with this.
If you are not extremely comfortable with them sleeping in the run I'd lock them all in the coop after dark. The coop needs to be dark enough that they can't see to attack each other. When I do this I just toss them in on the coop floor and let them sort out where they sleep in the dark. Some people like to set them on the roost but I don't see any real benefit in that.
Then I'd be down there at daybreak when they first wake up to see how it is going. If they are trapped in a small space together and it turns violent there is no room to run away. It can get serious pretty quickly. What I typically find when I do this with chicks and adults is that the adults are on the coop floor while the immature chicks are up on the roost or hiding under something. They are avoiding the adults. This is where clutter is important. It allows the ones that need to hide from the others to hide or avoid. Your Silkies can't fly so they can't go to the safety of the roosts if your roosts are high enough to be a safe place.
One reason I don't see a big benefit of putting them on the roost is that at that age your chicks may not be roosting anyway. With my chicks being integrated with adults they are not going to be sharing the roosts with the adults. Yours are close enough together they might if they can get to the roosts. Or they might decide to sleep in one huddle on the coop floor. I don't try to force them together but let them work that out at their pace.
My goal in integration is that no one gets hurt. All that other stuff of them becoming one happy flock hanging together and sleeping together can come later. It will as they mature. That's why I don't care where mine sleep as long as it is predator safe and not in my nests.
I've never locked strangers in the coop at night and expect them to wake up in the morning as best friends. My brooder is in the coop so the chicks grow up with the flock. My method of integrating them is to open the brooder door and walk away, usually when the chicks are 5 weeks old. I let them sort things out. If the main coop is crowded I raise chicks in my grow-out coop when I remove them from that brooder in the coop. In that case I let my chicks roam with the adults and the rest of the flock for a few weeks before I lock them in the coop at night to train them to sleep in the main coop, usually when around 12 weeks old. Yours are not total strangers. As long as your coop is a reasonable size with a few places to hide I would not expect that many problems even if your age difference was greater. But you never know what will happen.
Good luck!