I'm sure that it is because she's nervous about the dog. After all, Lucy was the same way when I first got her. However I can't just grab her and take her to a different room like I could with Lucy. Lucy, while she hates being grabbed, will at least tolerate it when she's being held. Alex on the other hand, not so much... It could be that Lucy might be more used to it, although I think I ruined it for Alex when I had to grab and hold her in order to wash dried baby food out of her feathers when she was still being weaned.
So when I hold her she freaks out. She'll scream and squirm, and once you let her go she won't want to go near you for days. Given that as a baby she was perfectly fine with it up until I had to scrub weeks worth of dried baby food out of her feathers, it's possible that being held had already been ruined for her (the funny thing is that I didn't even do anything to hurt her while getting the baby food out. All I did was gently soak the feathers until the chunks of crusty baby food would fall off).
So it's a bit of a catch 22 situation here. Try to let her out on her own and she's afraid the dog might get her. However try and get her out of the cage and she'll be afraid of your hand.
However we have made some progress today. I let both of the dogs outside and decided that I'd leave the cage door open while I'd get the birds fresh food and water. Alex, after being curious enough, finally decided to step up for me... Then she climbed to my shoulder and tried to fly away.
Her feathers finally grew back so she can fly a little further than usual, but she's still gonna need practice. She flies kinda lopsided given that she's used to using one wing more than the other. She appears to realize her feathers are back and that she can fly a little better now though, which gave her enough confidence to try and fly into the kitchen while I was cleaning out one of the water bowls. Granted she flew to the table, then into the kitchen and landed on the floor, but given that she got that far then stepped up and let me get her back to the play stand without flying away from me in fear that I'll grab her... I'd call it progress.
For a while she didn't want me to pet her while out of the cage. Wouldn't want to go back in either when it was time to go back in the cage.
Then at one point I was just humming, singing and whistling some random tunes in my head. They always pay close attention whenever I'm doing one of those three things. So for the sake of it, I decided to hum the tune I always do to calm them down whenever they're startled. Lo and behold they both started preening and fluffing their feathers, and Alex both stepped up for me and let me give her scritches. Lucy on the other hand only wanted the scritches and had no desire to step up. She'd just lower her head whenever I put my hand there to ask her, and no amount of scritches would satisfy her enough to agree to stepping up.
I think what was different this time is that she saw that the dogs were outside and couldn't get to her. The previous times, we'd just put the dogs in a different room, so Alex didn't know if they weren't actually just hiding. The only problem is that it's really hot outside, and even with a bowl of water, it wouldn't be safe to keep them out there for more than an hour... Maybe two.
Luckily it's starting to (ever so slightly) cool down. By around early-mid October I think, it shouldn't be as much of a problem to keep them outside for a bit.