It does sound like these behaviors could be minimized (in comparison to where they are right now) with the proper patience, enrichment and behavioral training, but they definitely are a ton of work and they are messy and destructive...Reducing attention-screaming means waiting out multi-hour scream fests that could last for a long time and get worse before they get better. The screaming thing is a common problem and can get super bad unless you respond to it properly--- either way, it's a process and will never go away completely, but can be decreased substantially with ABA (Applied Behavior Analysis).
1. Is your parrot getting 10 hours sleep on a schedule nightly? Under 10 messes with their hormones, mood and immunity.
2. How is diet? High sugar or carb diets can amplify behaviors.
3. Do people enter the room, look at the bird, talk to the bird or react in any way (whatsoever) to the screaming? Reactions= talking about it, saying the birds name, changing human behavior to compensate for screaming (e.g., yelling to be heard while the bird is screaming, moving your bird, looking at your bird, entering the room while the screaming is still happening, moving closer to the bird etc?) If anyone enters the room who wasn't already there (during a scream fest) the bird is going to take that as a win ASSUMING the screaming is attention-seeking...but before assuming that, you have to make sure sleep, diet, hormones etc are all in check.
4. When you pet, do you stick to head and neck only? If not, you should, as petting elsewhere is a sexual/hormonal trigger.
5. Does your bird frequently access shadowy spaces like huts, tents, boxes, drawers, bedding, under furniture etc? If so, stop allowing that access, as shadowy spaces are a major hormonal trigger (as is petting anywhere other than the head and neck) and with hormones, you end up getting behavioral issues..which is why you have to make sure basic needs are being met before jumping into behavior analysis.
6. How much time does he spend out of his cage daily?
7. Does he have toys that he uses? If so, what kind? They do need to be taught to play with toys and are often scared of new ones, so a gradual introduction (before just plopping them in the cage) is important for some birds.
8. How big is his cage? A cage that is too small or lacking in stimulation can also lead to screaming.
9. When you leave the room and as you do things around the house, are you using key words and phrases so that your bird learns to associate words with routine and can better anticipate your return?
10. Before the screaming starts, do you try to prevent it by talking to your bird room-to-room. If not, you should. Never respond once the screaming starts and when it stops, make sure you do a slow count to 10 or so before re-entering...In general, all bad behavior (unless you know the exact cause) should be ignored. If screaming is happening, you shouldn't come back into the room or provide attention until it stops for a set period of time. Let's say you are in the room and bird starts screaming at you. Don't look at your bird or speak to it, don't approach etc. I would walk out of the room, or stay in there with my back turned (with earplugs) until the screaming stops for a solid 10 seconds. If that is too hard, try 8 to start. The second the screaming begins, wait for it to stop. You will then count in your head (1 Mississippi, 2 Mississippi, 3 Mississippi... until 10). Any screaming that happens before 10 seconds will restart your silent count--- so, you have to start back at 1 the next time there is a break in the screaming and don't attend to your bird until it stops for the set period of time. At 10 s of silence, come in and praise your bird for being quiet using a quiet voice. No new people should enter the room during screaming and those who cannot ignore it should leave the room rather than reacting. This only applies to attention screaming, not fear or screaming from pain/illness (which is totally different). If you decide this is attention seeking (after correcting any issues with diet, sleep, cage size, hormones etc) then make sure that everyone is 100% on-board---if you leave your cell-phone or keys in the bird room and he starts screaming, you must wait it out until he is silent for 10 s, because entering during the scream-fest will reward an attention-seeker (even if your reason for entering was totally unrelated to the screaming itself).
11. If your bird makes a sound other than a scream to get your attention, do you make it a point of providing immediate attention for more preferred sounds? If not, you should.
12. Have you tried station training to cut back on destruction? You can train them to stay in an assigned area (although it's not fool-proof, obviously). That is another reason to have portable play-stands.
12b-If you don't want him chewing furniture, have you made it a point to introduce him to a variety of acceptable chewing options (in conjunction with station training)?
13. Do you have multiple (or portable) play-stands in the home so that he has a designated spaces with thing to chew of his own in multiple rooms?
14. Do you ever leave for super short periods and then re-enter right BEFORE the screaming starts and reward the quiet (using the same phrase, like, "good waiting" or "thanks for staying quiet". I tell my bird "wait" etc when I hear her talking from the other room. She almost always talks before getting to the point of screaming because I praise and attend to any talking but ignore screaming. Ive worked with her a lot on it for years, but these days, when I step away and she is feeling social, this is the progression of what I hear: "hi", "hello baby", "big bird", "I love you", "wanna look outside", "hop up", "come here", "COME HERE", "BACK BACK BACK", "AHHAHHAHHHHAHHHAHHH!!!!!" If I can respond or re-enter BEFORE it gets to the screaming, then it rewards the preferred vocalizations and prevents having to wait out a scream-session with the counting trick etc.
15. Birds are flock animals and need a ton of interaction. If you are keeping him in your room or excluding him from family (flock) activities, he is going to scream to try and get your attention. They need at least 3 hours out of the cage and at least an hour or so of really focused/direct interaction (involving shared games, puzzles etc-- some of this time can be petting, but you don't want to make petting your primary form of interaction, as it can become sexual in a captive parrot if your only direct interaction is petting. You should try and get the whole family involved and keep him in the hub of the home. If the hub is too noisy for him to sleep, a smaller sleep cage (or another cage in general that can fit in a sleep space) is advised-- one that can be placed in a quieter/darker area. If your bird is covered in the main area of the house and people are still talking/laughing/ shuffling things around or walking by the cage, your bird may be quiet, but it is very unlikely that he is actually sleeping.
If you re-home, I would just be very careful about who you give him to. If you do it without an adoption/foster/rescue, I would make sure that the person you re-home him to has a lot of experience with parrots, and if they send you videos or pictures of their home, I would ask them to hold a sign with a code-phrase or something on it so that you know it is actually them and not someone using someone else's images/videos. If they currently have other birds, hopefully they are already aware of the importance of a 45 day quarantine. Reverse image searches are also hepful in determining whether a photo belongs to someone, as opposed to having been recycled from someone else's posts/images on the internet.
If you re-home him to the right person, I am sure he will be okay. I won't say that re-homing isn't hard on them, but as you said, if you can't provide him the life he needs, then re-homing may be for the best. My U2 had 3 homes before me and she's only 14--it took her a bit to come around, but she is pretty happy with me even though she was with one of her homes for over 4 years.