Welcome to the forums! You have a real cutie there and you've come to the right place to help steer you in a better direction.
In general with animal training, you need to teach them what TO do, not what you want them to stop.
Parrots very rarely bite in their natural flocks. They don't "randomly" bite at all. I think you need to go back to basics, and start over with building trust and repairing your relationship.
Start over entirely - don't think about the things you don't like. Pretend you just brought home a rescue who wasn't treated well in his past home and that it's only been a few days. How well do you know his personality? What is his favorite thing to do with you? What are his favorite foods, his favorite toys? What kind of training activities are his favorites to learn? What times of day is he most chatty? What are your favorite things about him?
Parrots are extremely intelligent and social and they need just as much mental stimulation as they do physical play. I've learned on this forum that many GCC use their beaks as part of their communication. As a team, you can learn together how to communicate with each other in ways that aren't going to be harmful - but it starts with you; you're the predator animal. Grabbing and shaking a bird is terrifying! He won't show how scared he is because showing fear almost guarantees they'll get eaten. Try to think about it from his perspective - he knows you are big and scary and can hurt or kill him, and so far, your actions are just proving that you're big and scary.
Write down the times he tends to go after you or your mom. If there is a pattern, don't have him in the room while you are doing those activities. You need to prevent the chance for him to bite. Read the suggestions about body language, and try to see if you can figure out what is happening right before he bites, then change the activities so that he's not practicing that same set of events.
As far as training, I'd recommend target training. It's hands off, and gives you something to do together that can be really fun. It also can be a way to gradually teach him to use his beak more gently. Here's a link to a post that spells out the basic idea:
Target Training
It will take some hard work for you and also the rest of your family, and you'll have to be consistent about being positive trainers instead of scary predators, but green cheeks can be the biggest cuddle bugs and most excellent companions once you do establish a good relationship!
You can also search the forum for other threads about bulding trust, bonding, and training - it's the magnifying glass up top.
Good luck and keep us posted