Edit: typos. So many typos.
All my parrots are rescues; bongo bounced through four homes before i got her from her previous owner. Tiny cage, wrong diet, grit which is not good for a green cheek conure, etc.
I recently adopted out two cats of five from an unexpected rescue i got asked to do for cats abandoned at a derelict farm with kittens.
It. Took. MONTHS. I got so overwhelmed trying to keep up with cleaning litter boxes for five cats once all the kittens were grown up with their mommas. I got very lucky; i finally found a home for Mittens, but it came right when i had asked to surrender her to a shelter when they had an opening to take her. I just couldn't keep up.
Most people who contacted me about mittens didn't even want her, they wanted the kittens or were trying to scam me for money.
I also turned down at least two people who ignored everything i so carefully wrote out about her, which included telling people Mittens could not safely be trusted with tiny children as she does bite when she gets needy, something that i was working with her on and was improving but not enough I'd trust her with a tiny kiddo. She wouldn't mean to hurt them, but she easily could.
No, ma'am, i will not give you this cat for your four year old son!
What I'm getting at; when I finally did surrender Mittens, i had to let go. I had the owner sign a contract saying they could not sell, gift, surrender, or rehome her in any way shape or form without my written consent first. I also had two other people sign as wishes to the contract.
The contact included her contact info and home address.
I checked in at the two week mark and got sent the most adorable pictures of the happiest cat ever.
However... Realistically, you can't guarantee whoever you rehome an animal to will follow through on the care you want. You can only use your best judgement and carefully vett all potential adopters. I've heard endless stories of rescues who have issues getting their contracts enforced let alone finding the person if they decide to move and not tell you, or if they rehomed the animal without permission and now who knows where they are.
The contract, the way i see it, is better insurance than nothing... And anyone who balks at the idea of the contract, isn't the right person you're looking for.
Also, having a rehoming fee immediately helps weed out a lot of people who just want a free bird. I actually listed Mittens with an adoption fee that could be waived if the person got her spay and vaccines and chipping covered, which this adopter did. She had the best clinic call to confirm with me that she did indeed schedule the required appointments.
Speaking of, if you can meet the vet the person adopting will be using, that can help a lot for worries.
You don't want someone who shows up and says they want the bird, no questions asked.
You want someone who wants to know EVERYTHING about the parrot. They're going to ask you how old they are, their history, what you feed them, if they know any tricks or commands, if they are "hand tame" and what any special quirks they may have. They will ask how the bird does with going to the vet, car travel, who their current vet is, they will ask you what the birds favorite and least favorite foods are, what activities the bird likes and doesn't like.
They will ask what training you have already done, if any, to work with the bird on their behavioral struggles and what methods have and haven't seemed to work.
You want someone who is ravenously curious about this parrot who needs a good match.
If you were in the state of Minnesota I'd probably be willing to try working with them for behavioral training; all the parrots i have brought home have been the "unadoptable" birds who bite and had other issues. All of them adjusted in time and learned to trust again, in their own ways.
That, incidentally, is your alternative to rehoming the parrot. Because your fears are valid; I'm Bongo's fifth home not including her time at the breeder, dispensary, and petstore, and Echo's second. My first parrot, rest in peace, was with his first home for like fourteen years before i adopted him. I had to teach him how to FLY.
In Echo's case, his owner adopted him then depending on if i go with the pet store employee story or the rehomer's story, she either rescued him from a bad place and he's him only to not be able to handle his special needs because her 
Amazon hated him, or she brought him home and then he got injured and chaos happened.
Either way... She was someone who committed to a bird only to rehome him mere months after buying. She meant well. But she couldn't handle his needs (he had/has a major fear of being confined and would injure himself in escape attempts. He has come a LONG way since we got him) because of the little flock she already had, and his arrival upset the other two birds.
But
The thing that kills me
Is both people LOVED these two birds i have now. They were unwilling / possibly unable, to work with them to correct the behavioral issues the birds had. Which can be done!
It takes a bucket ton of time and effort and lots of studying and learning and trying new techniques because what works with one bird might be catastrophic with another.
Bongo took me three years to rehab. It was a full year before i let anyone else try to hold her-- i got bit bloody every. Single. Time. I. Handled. Her.. Now i get bit maybe a few times in an entire year, and mostly only during breeding season when she's very cranky.
 Echo took two weeks of constant effort, then we had him comfortable in his cage and going into it on his own. Echo also only had about six months of trauma, whereas bongo was not quite five years old when i asked her, and had been with her neglectful owner for at least one to two years. That's seven homes (her perspective) in not even five years, during her most crucial time of development (baby to and through puberty). She was one messed up birdie.
If you really, really, really don't want to rehome the conure... Try instead to find a parrot teacher who can help you train them. Except in species and language, they are no different than rehabilitating an abused horse, dog, or cat. With enough time and lots of learning, you can teach them new manners. That respond incredibly well to positive reinforcement, consistency, and routines. A note of caution on that, they can be over trained to a routine so bad that changing it then stresses them. I purposefully change things up a bit every day to keep them comfortable with change.
Echo doesn't want to be touched or held but he will listen to most my directional commands now. Bongo has gone from being a very dangerous bird to being the little love bug of the house i can hand over to most any stranger and she'll sing to them and maybe even cuddle up.
So, i hope that gives you hope for another option, if you are unable to find someone you trust to take in the conure or decide you'd really like to keep her.
I wish you both luck whichever path is taken 
