Welcome and thank you for recognizing a less than ideal situation and being willing to take this sweet baby on. Blue fronts are great birds, very intelligent, very capable of forming strong bonds and females tend to be the 'easier' gender temperament wise. My BFA isn't a talker either, though I *wish* he'd make those beautiful 'opera noises' you describe. Those are normal amazon noises and very melodic and beautiful

A few things to address your concerns/questions:
A 27 year old cage that has been exposed to the elements all that time? I cannot even imagine the condition. With the exception of stainless, cages generally need replacement every 10 or so years. So unless it is a stainless steel cage (in which case it is absolutely worth cleaning), it's time for a new cage! Don't even bother washing her old one, have a nice new one all set up for her upon arrival and take the other one straight to a scrap metal place for recycling. Don't worry about her having never had a different cage, she's going to be experiencing a lot of changes joining your home and it's best everything 'new' happens at once IMO. Get them used to the new flow of things right away. This is the cage my blue front has and it's a great size for a large species amazon with lots of room, though there are many different styles of cage to choose from if this one doesn't appeal,(dome top, flat top, top with a play-gym etc...) but do try to keep any cage you choose in this general size range. Remember, BFAs are large, active parrots and need a lot of room to be active! Even one who has a lot of out of cage freedom will still likely spend a lot of time in their cage.
https://birdcages4less.com/page/B/PROD/PA5710
I also recommend just biting the bullet and investing in a bird backpack now, as eventually you'll be wanting to bring her out on walks or to the park with you and your dogs, running errands etc... (all VERY good activities for a birds mental health), but if cost is a major consideration with all the supplies you'll be needing, any small cage will do for strictly transporting her home/to the vet etc... My mom transports her amazons to the vet in a "parakeet" cage. As long as they fit comfortably through the door and can physically turn around in it, a transport cage can be small since they won't spend much time in it. Just be sure if you get a small cage to put an appropriately thick perch in it for an amazon. This is the bird backpack we have and it is AWESOME to get my bird out and about in and buckles into the car easily (also never any problem carrying him on public transport in it either). This is our 4th year with it and it's holding up great:
https://www.celltei.com/products/pak-o-bird-modified-medium-for-congo-african-greys.html They also have a newer and more affordable option, though I'm not 100% sure on sizing for an amazon so you'd probably want to contact the company and check:
https://www.celltei.com/products/birdie-gogo.html
As for actually getting her in a transport carrier/cage? Have the staff move the cage into some kind of enclosed, indoor room or space so she doesn't fly away the second that outdoor cage is opened and someone gets in there trying to get her. It is possible she'll step up and it'll be no issue but it's equally possible she'll be scared out of her mind and make a break for it. Have that happen in an indoor space! Fist option would be to see if she steps up for anyone at the place and have them move her from old cage to travel carrier. If not, try seeing if she'll step up to a stick. Get a 1 1/2" thick dowel from any hardware store and cut it down to roughly the length of your arm to the elbow (this will also be important to have once she's home, as you train her). See if she'll step on it for transfer, lots of infrequently handled birds are less frightened of a stick than a hand. If not, try enticing her into the carrier with a tasty treat. No luck? Last resort will be to bring an old towel with you, not too big and preferably of a neutral color. Brush up on how to towel a bird because it may be the only option to get her into the transport. She will forgive you, eventually

Be as gentle and calm as possible during the process. They feed off our vibes, do not be scared or she'll be scared! ALL of these options MUST be tried indoors only!
Our amazon is totally "free range" but still spends a good chunk of his day in his cage of his own choosing (even closes his little bird door behind himself

). If no one will ever be in the bird room except to take care of her, have a play stand or something in the family room so she can join you. If you get a backpack, that would be suitable too to protect her from the dogs if you can't kennel or put them outside/another room for her safety, but she needs somewhere she can safely hang out around her new family. She may be shy at first, but if you do it right, she won't be shy for long! Amazons are VERY social and like to hang out with their human "flock". Once they get to know/trust you, they do not like isolation and really want to be in the middle of the action all the time. Be sure you set things up to be conducive to her being able to be around when everyone else is home and hanging out

She can have her independent play time when everyone is at work/school or otherwise busy during the day.
Food wise, I just feed my bird on primarily human food (prepared specially for him so no salt, sugar etc...). Much cheaper than "parrot food", no worrying about the low standards for pet food safety VS knowing human food safety is taken MUCH more seriously. I'd argue it is healthier and more interesting for him as well (imagine eating vitamin fortified, unflavored, dry cereal every meal, which is effectively what pellets are. Blech!). Human grade beans, grains, fresh produce and nuts... comprise about 80+% of my birds diet. He also gets a high quality seed mix for daytime foraging. I do not leave a big dish of food for my bird all day (though he always has a water dish). He gets a morning and evening meal, each removed after he's finished eating. I portion to be just a little more than I expect him to eat so we have little food waste (parrots do NOT need a half full dish of food, it's just WAY more than they can eat at a setting!). He gets treats to forage during the day, but just enough to be a snack and not for him to fill up on. This encourages him to eat everything provided in his meals, as amazons can be picky eaters and mine is the president of the picky eaters club. Vet says he's in excellent health for his age (about 20). My mom's been feeding her 40+ year old parrots in a similar fashion all their lives and they're both in excellent health for their ages and eat just about anything you give them. There are some foods that are toxic to parrots, like avocados, onions, mushrooms, chocolate etc... Familiarize yourself with bird-safe and toxic to birds foods.
Speaking of food, if you have ANY non-stick cookware, bakeware, small appliances, oven liners etc... they MUST be gotten rid of and replaced with PTFE/PFOA free or your home is not safe for a bird. Doesn't matter if she's in another room, "teflon toxicosis" is real and kills thousands of birds every year who's owners thought they'd be the "exception" who could keep using non-stick for whatever reason. The fumes are odorless, quickly spread through an entire house and do not affect humans or dogs or other larger pets. Parrots simply drop dead from it and often after you've been cooking with non-stick for a while around them. Just takes one time for it to overheat, go in too hot of an over etc.. and the bird is dead in minutes. NOT worth it. If you can't give up non-stick, you aren't the right home for a bird.
Lastly, line up an appointment with an avian specialized vet. You're dogs vet may "see" birds but is simply not experienced enough. Find an avian specialist. I wouldn't say theres a particular rush since she sounds to be in generally good health but it's a good idea to have any new bird get a wellness exam and in as a patient in case you do have an emergency. I'd say within a few weeks of her being home, she should visit the vet.
Hope this all helps and best of luck! I think you're going to love having your new avian overlord

