Welcome to the forum and thank you for adopting instead of buying! First of all, what's his name and do you know for a fact it's a male?
Now, your problem is VERY common with GCCs. They are either the sweetest birds or little mean pitbull birdies

The problem with them is that they are EXTREMELY needy. As you already noticed, they want nothing else but to be on their human. To them, having their body touching yours is not a luxury, it's a necessity; and people, having normal lives, cannot provide this -thus, they end screamers and biters. And, because they are still needy, one finds oneself between a rock and hard place! What to do? Allow him to be on one's shoulder and get bit or put him in the cage, making things worse? It's a quandary, isn't it?
Well, I tell you, I've had four (have only one right now, a female), all four of them were given up because of 'behavioral' issues (biting, screaming), of them, the one that took the longest was one that had been abused by his first owner (this is according to his second owner, sister to the first owner) and he was just like you describe: he wanted to be on me, especially burrowing down my top (which made me end up getting bit in 'delicate' places

) but he bit me all the time. Unfortunately, the ONLY way to 'cure' this is to prove to him that he will get what he needs -and that means putting up with some bites.
Wolf is right in that the best course of action is to try to prevent him from biting you -this is not only because nobody likes to get bit but also because, although you don't want him to feel too bad about it (it's not really his fault), you still don't want to get him used to doing this all the time. Parrots only bite as a last resort. Pet parrots do it out of fear, to 'defend' their nest (cage) or 'mate' (chosen human), when they have no other way of getting their point across or, as in his case, out of sheer despondency. Most parrots are very forgiving but the needier they are, the more they will 'act up' and GCCs are VERY needy (I think as much as any cockatoo). People don't realize this and because they are small and cute, they think they are what people call a 'beginner's' bird which, in their minds, means 'easy to keep' so very young, very old, very busy people get them and these are the ones that, unfortunately, should not be getting extra needy birds for the simple reason that they cannot make them happy.
Now, there are things that help. A good diet is essential because these are birds that eat mostly fruit in the wild so free-feeding any type of protein food (seeds, nuts, pellets, nutriberries, avicakes, etc) is a no-no. They need a fresh food diet - things like gloop, chop or mash accompanied by fresh produce (especially fruit which should be given in portions larger than one would think a little bird would eat) and a small portion of a good quality seed mix for dinner. A solar schedule helps A LOT because this ensures that the bird is not hormonal all year round (sexual hormones = aggression, especially in males). Routine is another one, knowing exactly what will come and when gives them a sense of control over their lives and reduces anxiety (if they know that they are only going to be in their cage for, say, a couple of hours during the day, they don't complain). But, with GCCs, what really works is having them on your shoulder for hours and hours because it's what makes them happy and a happy bird is a 'good' bird. There are times during the day when birds are more prone to interaction and some that they rest.
You can wear long sleeves, something draped around your neck (to prevent him from biting it), a wool cap (to protect your ears) but don't wear gloves because it will make things worse (they HATE gloves). Always have something handy for him to chew on (a little stuffed toy, like Wolf suggested is good but experiment with different things like a untreated sisal rope knot, for example). Put things in his cage that he will like, like a little sleeping hut or a tunnel (they like hidey holes). Make him fly as much as possible because a tired bird is less likely to bite than one that has 'ants in his pants'. Try to get him to take a good bath every morning (they love baths!) and it's a good way of distracting them, get them tired and entertained for a while because they have to preen after the bath. See if you can get him to target train - but you will need to get him on the fresh diet first because you will need what it's called a 'high value' item as reward and these are always some sort of high protein/fat food like a sunflower seed (but it's better if you offer something like an almond sliver or a piece of walnut).
For example, you can uncover (if the cage is in a human area or in front of a window where he would get the light of a street lamp) his cage when the sky is beginning to get lit (at 6:30 am this time of the year) and allow him to come out on his own (they are usually pretty mellow when they first wake up) and you can clean his cage, have a cup of coffee, etc (but you can't turn on the artificial lights, it has to be done with the natural dawn light), then, when the sky is completely lit (around 8:15 am) you turn on the artificial lights, put his breakfast in his cage and put him back in it. Half an hour later, you open the door to the cage again and offer him a bath (this takes about 20 minutes from beginning to end because they need to preen after). Then you can put him on your shoulder and do whatever for a couple of hours. At about 11:30 am, you put him on a playstand or on top of his cage and do target training for 10 minutes, put him back in his cage and give him a treat (always give him a treat when he goes back to his cage). He should then rest for 1 to 2 hours. Then you open the door to the cage again and let him out to play with you -you can put him on top of a table while you sit in front of it and give him little things to play with for a while. Then you can put him back on your shoulder for another couple of hours. And, by 4:00 pm, you turn off the artificial lights and put him back in his cage with his dinner. Once he is fast asleep, you cover his cage so he can go to sleep (again, if the cage is in the middle of a commonly used human area, you will have to keep the noise down so he can sleep).
See what I mean? I know it's a lot of work but it's pretty much the only way things work with parrots... some are more independent than others but GCCs are not and, when you are dealing with one that was obviously deprived of the attention he needed, you need to make up for it. But things do get easier as they begin to trust you and realize that their life is now good.