I've had my share of nervous, unsocialized animals. One of our dogs, Abby, is a little 3lb chihuahua. For the first two years or so, she wouldn't let us pick her up. Instead, she'd run to her bed or crate to be picked up. She was very skittish, and wouldn't let any of our guests touch her. Now she's a little flirt and loves having people hold her and pet her.
Gizmo, a 7lb long-haired chihuahua, was abused at his former home. His human ran a daycare, and the kids would hit him, carry him by the throat, and she was worried her young son would seriously injure him. For the first few months, Gizmo constantly thought we were going to beat him, and he was really scared. If we were carrying a newspaper, broom, etc., he'd run in fear because he thought he was going to be beaten (we wonder if the husband had abused him).
One evening, I was babytalking him and telling him he was a good little boy, and he accidentally wet himself. You should have seen him. One moment he was really happy and was opening up to me, the next he had a look of shock and fear as he cowered, trying to get away from me. He wasn't just upset because he'd accidentally peed in the house. In that moment, I'm sure he thought I was going to yell at him and hit him. I told him that it was okay and that he was a good boy, but he was terrified and wouldn't come over to me. I had to pick him up and cradle him and reassure him until he calmed down. Today, he's still a dork, but he's a lot more confident and isn't constantly worrying that we're going to abuse him.
Alice was only 2 or 3 months old when I adopted her. She'd been rescued from an animal hoarder who was going to "release her into the wild" in the middle of the winter. Alice had bad PTSD and was completely withdrawn and was constantly in a state of fear. The other budgies (except Ziggy) avoided her because they could tell there was something wrong with her, but after a year, she finally got past her PTSD and explores her surroundings, socializes a bit, and is actually enjoying life.
With Sol, I'd take things slow with her. Sit at a distance from her cage where she still feels comfortable enough to preen, eat, and sleep in front of you. Hang some spray millet in her cage and read to her. Don't stare at her, and keep one eye closed when you look at her. Keep these sessions short at first. 5 minutes, for example. As she gets more comfortable with you, you can pause for 15 seconds or so and give her the slow blink and maybe grind your teeth. Blinking slowly and keeping your eyes closed for a few seconds each time shows her that you're calm and that you trust her, and that you don't want to hurt her. If she starts slowly blinking back and copying you, awesome! That means she's opening up to you and is willing to let down her guard a bit.
When you have to give her fresh food or water or whatever, gently tell her what you're doing and make slow, but deliberate motions. Once you're done, put some sunflower seeds in her dish and walk out of the room, giving her time to come out of her Happy Hut and eat the treats.
As she becomes more and more confident in your presence, move your reading chair a couple of inches closer to her cage, and read for a bit longer. If she feels uncomfortable, move your chair away. Once she's used to this bonding routine, try reading a kid's book with pictures in it, and gently show her the pictures at a safe distance. Talk to her about the pictures and comment on what's happening in the story so far. Is one of the characters being mean? Talk to her about it and try to engage her in the story. If she starts becoming interested in story time, that's fantastic! Even if it means she's peaking her head out of her hiding spot to take a look at the book when she doesn't think you'll notice.
Again, she dictates the pace. At no point should she feel unsafe. Slowly, she should become more confident, knowing you respect her privacy and won't force her to do anything she's not comfortable with.
When you're in the room, make sure there's always some noise, such as gentle talking, humming, or soothing music. To parrots, silence means danger, and a quiet human is a predator.
