Michael, your comments seem unnecessarily harsh. I appreciate your creating this forum and I enjoy the videos you post on YouTube, and I feel sometimes the things you say to people here border on the mean. Telling someone essentially "you f-d up, now live with it" isn't helpful. To borrow a line from an episode of Buffy the Vampire Slayer: For someone who studies human behavior, you could try showing some.
Annne, understand how difficult it is for people who aren't able to visit and assess your situation to try to offer diagnoses and treatment for parrot behavior problems on line. Since we aren't there, it is incredibly difficult to tell what is actually going on and the best we can offer are generalities and inferences from our experience with our own birds.
When I read your original post, I definitely got the feeling that you may be doing something to reinforce your conure's screaming. Without being there, it is impossible to tell if that's the case or to offer specific suggestions. Companion parrot behavior problems are going to fall into one of three categories: natural behaviors that are maladaptive to living in a human home, neurotic or illness caused behaviors, or behaviors that are being reinforced by the bird's humans. What you described in your original post could be any of those things. I don't know what.
Here's what I do know: conures are flock animals who want a lot of social interaction; sun conures in particular are notoriously loud screamers; parrots are extremely smart animals and they are capable of learning new behaviors.
My Nikko screams for a couple of reasons. First, if someone he doesn't know is in sight, he will scream as an alarm. That rarely happens. Second, he sometimes just seems to take joy in being noisy; that's a conure for you. Third, and most frequent, he screams when he wants my attention. In your case, we can rule out alarm calls. It also sounds like you're dealing with more than the usual conure vocalness, so we're left with the third reason: attention.
Nikko has two types of screams. The first is a pair of two short, loud calls. I've learned that this is a contact call. He gives them when I'm out of sight and he wants to know where I am. This is an absolutely normal, expected behavior, so whenever he gives these contact calls, I respond by saying, "I'm right here" in a normal voice, wherever I am in the house. We'll sometimes do this a few times, then he usually moves on to other vocalizations. (Dante has also taken to responding to Nikko's contact calls with "I'm right here" LOL)
The second scream is an extended series of short, piercing calls. I imagine what your sun conure is doing is something like this. Nikko does this when he wants attention. He is more prone do this vocalization when he is molting and right after a time change like we just had going back to daylight saving time. I don't know how natural this call is in the wild, but it is what I consider "maladaptive" for living with people. When Nikko screams like this, he is expressing a real need, but doing so in a way that isn't appropriate for my home. I don't respond when he screams this way. Note what I mean when I say "I don't respond": I keep on doing whatever it was that I was doing in the first place. If I was being quiet and reading or watching TV, I keep being quiet. If I was talking with someone else, or engaged in some other vocalization of my own, I CONTINUE IT. Stopping and being quiet is a response and birds are smart enough to put two and two together and realize they can make you stop talking by screaming. That is a form of attention and can reinforce the screaming behavior.
While this may be "maladaptive", it is more or less natural and it isn't going to go away completely. That's why I taught Nikko to say "I love you" -- he can use that as an acceptable substitute behavior to screaming for attention. It is MUCH easier to replace an unwanted behavior with an acceptable variation than it is to just make the bad behavior go away. When Nikko wants attention, he knows screaming won't get him any, but saying "I love you" will, so he is much more likely to say "I love you" than he is to scream. Try to find a less offensive, or possibly even pleasant, vocalization that your bird does, and reward when it happens while not responding at all to the unwanted screaming and see if you can redirect your birds behavior.
Finally, your description of your bird flying around and nibbling on you and your partner makes it sound like your bird is really out of control. My boys are not allowed off of their cages unless *I* take them off myself. I know this won't be a popular suggestion on this board, but you should really consider clipping your bird's flight feathers. There are probably good arguments to be made for allowing a bird to be flighted, but it sounds like your bird's ability to fly around at will is damaging your relationship. Clip the wings and your bird will learn if it wants off the cage, it will have to step up without biting. It will learn that if it misbehaves, it can be put in timeout and restricted to its play gym. Having a flighted, untrained bird loose in the house is not a great idea. You can always let the feathers grow back once you can trust the bird to behave itself. At the very least, you should try CONSISTENT, in-cage target training with your clicker. Being out of the cage is a privilege, not a right.
And finally, finally

while I don't want to place any blame on you, I will place responsibility on you. Our birds' behaviors are largely in response to things we are doing. You need to be constantly asking, not "why did my conure do this?" but "what did I do to prompt this behavior?" Have you ever watched "The Dog Whisperer" program? Cesar Milan rehabilitates dogs with behavior problems, but he TRAINS the dogs' owners. It is nearly always a case that the dog's problems are caused by the owner's behavior. I believe the same thing is true with parrots. (Milan also demonstrates that you can, sometimes easily, "untrain" bad behavior, in spite of Michael's assertion to the contrary.) Parrots are not people, and don't respond to human motivations. You need to learn to think like a parrot.
Your original post sounded like you are very distressed by your conure's screaming. I can definitely understand that. I will say I admire you for sticking in there and for your commitment to your bird. I hope you are able to find a solution.