Hi there! Welcome to TF!
It sounds as though your pair is closely bonded to each other! Maybe you'll need to move even more slowly around them, and make sure you are below them, ie, have the cage quite high up so that they don't feel threatened by you. Do you spend a lot of time sitting by their cage, talking softly to them? I suppose you'll know all about step-up training and such, but it may pay off to "start fresh":
1. Sit by their cage, talking softly. They should calm down after a while.
2. Move your hands to the side of the cage, and leave them there while you talk or sing to them.
3. When they don't freak out anymore when you do this, move your hand slowly inside the cage and leave it near the opening. Sit like this for a few minutes every day, until the birds calm down.
4. Move your hand a little closer to the birds, step by step. Use a short perch to get them to step up, if they are still afraid of your hand.
It may take a long time for them to become comfortable with you around, but they should get used to you eventually.
The new budgie may assassinate the taming process a little, if they can hear each other. Have the couple ever been out of their cage? It may be worth letting them out in a bird-proof room to get some excersise. The screaming could be caused by frustration for not being able to fly, or boredom if they don't have toys that get changed around. (Are they in a large enough cage?) If you have a bird gym they could land on when out, then you could use the opportunity to clean their cage while they're not in their cage, and you won't frighten them as much. They'll find their way back in the cage when they get hungry, just make sure food is inside the cage only and not on the playgym. You can use treats like spray millet to assist in the taming process.
I hope you give them another chance. It is very rewarding to get budgies tame, but even if they never warm up to you, they'll still provide you with their song and beauty and quirky mannerisms....
I'd be keen to know if you have any more luck as time goes by!
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