I'm glad you guys took these 2 in, I always wonder why people buy these birds and then treat them the way they do...I can't imagine that this guy just bought these 2 Amazons 2 and 3 years ago and just threw them out like that...Don't be surprised if the Avian Vet tells you that they are older than 2 and 3, it's amazing what people getting rid of birds will tell people. I hope that the Vet you are taking them to is either a Certified Avian Vet or an Avian Specialist Vet? These guys need to see an actual Avian Vet who knows what they are doing and talking about, not an "exotics" Vet.
They should both have a complete wellness exam, as well as routine blood-word (baseline blood work), as they probably both have some liver disease simply due to the diet of only sunflower seeds. The blood-work will tell you if there is any infection, anemia, etc., and then also given their liver and kidney function values, which are very important in this case.
If the male is "very overweight", then don't be surprised if the reason he is walking around like he's in pain is due to severe arthritis in his joints from the excess weight. It could also be Fatty Liver Disease due to his diet, which causes the liver to become extremely enlarged. Then what happens is the huge liver starts pressing on other organs and can cause pain, as well as trouble breathing under exertion, due to the enlarged liver putting pressure on the air-sacs. I actually saw an Amazon not long ago that was fed an all-seed diet (seed-mix, not just sunflower seeds), which is also bad, and he was extremely overweight, and was limping. He also started panting when he would simply walk across a room. We took x-rays and his liver was so large it was pushing on his air sacs, as well as putting pressure on his entire chest cavity. His x-rays also showed severe arthritis, which was worst at his knees, then his ankles, then his hips. He basically had no knees left due to the weight he was carry around. This guy was a human doctor too, an Internist of all things, and yet he still only fed him a bagged seed-mix every day for 10 years. It was bad. And your female will most-likely also have the same liver issues, as she was eating the same diet as well. However, the reason the female isn't as overweight as the male is because she has been laying eggs, probably chronically. This is extremely hard on their bodies, and they tend to be both undernourished and malnourished, which these two birds are anyway due to their diet, so she's probably in pretty bad shape too. Make sure that she always has access to both a Cuttlebone and a Mineral Block, as she's most-definitely Calcium-deficient from the egg-laying. Hopefully once you get them settle-in you can work on eliminating the egg-laying, because over time this can kill them as easily as the Fatty Liver Disease can.
Let us know how it goes at the vet, I certainly hope the vet is an Avian Specialist with lots of experience, and knows what they are doing, and what tests need to be done. These guys need some help to get going, but it sounds like you've already got them on a great diet. Limit all fatty food or food high in sugar, like seeds (especially sunflower seeds and millet), nuts, people food, and even fruit, as it's full of sugar, and that sugar is turned into fat that also accumulates in the liver. Fresh fruit should be only a 2-3 times a week thing, not every day, and not in large quantities; fresh veggies are an every day part of their regular diet however.
I know it's going to be hard. I just want them happy healthy.
I'm not smart enough to add links. But some of the brainiacs are sure to. You might have good luck with warm cooked foods, like squash, broccoli, peas, carrots. Be careful with juice, I'm not a 100% sure but I think it's a no no, at least any more than one sip..
