Firstly, and I know that this is definitely a last resort for you, but exhaust every possible option (even waiting just a bit) to not put your bird up for adoption! She definitely loves you so much by that behaviour, and you do too clearly so I'm sorry that it's gotten to this point with her!
I truly hope everything gets better for you and Pixel, and I hope this helped!
- It's interesting that you got her a friend and there hasn't been much change, but just like anything with birds it takes a long time for them to change, so if you only get her a friend recently possibly wait it out, they may grow closer and not need you as much. She may also simply like you more. My cockatiel (while definitely attention-seeking) doesn't bother me too much because I've gotten used to birds screeching loudly around the house and there are lots of cockatoos in the forest I live near that come around and entertain my cockatiel.
- Also I'd be interested in how much out of cage time she gets. I totally understand that not everyone's situation allows for a bird to be out of their cage constantly or even part-time, but if possible definitely give them a chance to be out. Even if they have a spacious cage you'd be surprised the impact on mental health it can have on them just not being able to get out.
- My cockatiel when I first got him was super clingy and I couldn't go to the bathroom either, but the more I let him out he got better, and he also found a permanent place he liked on top of the cage, which meant he could stay there when I needed to do something.
- Cockatoos/cockatiels are extremely social birds. Many cockatoos live in gangs and have a chance to socialise constantly, and they scream loudly to mourn the death of a member. Loudness is just built into cockatiels (my one has been singing loudly the entire time writing this) and it's very hard to change that behaviour since you are a part of their flock, so they need social reinforcement from you. You're doing well by positive reinforcement for quietness. When she's squawking for your attention, try going up to her and if she starts singing or chirping instead (even if it is still a bit loud), reward her. She may not enjoy being completely quiet but she might enjoy chirping and singing if she gets a treat with it.
I truly hope everything gets better for you and Pixel, and I hope this helped!