Hi Chris,
He is very young. I don't know when ring necks typically wean so I hope he is eating for you. It can be very stressful for any baby to move homes and they need time to adjust. If they are also worried about food, the stress is even greater.
How and what is he eating?
My advice is likely not what you want to hear since you are anxious to touch him, but I think you should leave him alone. Just service his cage and talk softly to him. Talk to him the entire time you are nearby. When you are talking it will be reassuring to him because instinctually he knows a predator will sneak up quietly.
My suggestion is to place at least one perch near the door so he can move there when he is ready to interact with you. He will see you open the door and know that is how to exit the cage. When he is ready to come out he will begin sitting on that perch by the door.
It may take him SEVERAL weeks to settle in. I hope you can be patient and gain his trust during this time. Reaching in trying to pet him will not hurry the process; instead it may set things back.
Meanwhile, are you sure he is getting enough to eat? The very most important thing for a healthy bird is that they are eating a sufficient amount of species appropriate foods.