I'm just scared she'll get bored and unhappy while I'm not home since she's always with me in the apartment.
I'm not a "bird person", I don't know that much about birds, but I've picked up a fair bit about training dogs since I've had them for a long time and took them to training. This might not apply the same to birds, so if a "bird person" tells you I'm wrong, listen to them.
To encourage independence, you need to make the "separation times" normal and no big deal. If you ALWAYS have the dog at your side and only put them out when you have guests, or occasionally go out (or whatever) the dog interprets that as being excluded from his "pack" and can go completely mental with anxiety and upset at being "shunned". You need to just teach the dog that he can be out, with you, out alone, in alone, in with you, all the things you want him to be comfortable with.
Instead of always being with you, you could consider making "cage time" part of his day.
We need to supply our pets with their needs - food, company, activity. We don't have to amuse them 24/7. In the great outdoors, they don't have a succession of parrot toys to amuse themselves with. They might just sit on a tree, chat with a friend, pick at some fruit or seed, whatever they do wherever they live.
If he's not stuck in a cage alone all the time, he'll know that his cage time is just part of his day and you'll appear periodically to say "hi", give him some yummies, fetch him out for a game or a cuddle or a look around.
It depends on the bird how much stimulation/company/etc he or she needs. I just supply food, attention, room to move, stuff to look at and pick at, and if they seem healthy and aren't doing anything "crazy" that demonstrates anxiety or upset, I just don't worry about it. Taking them from their natural environment, sticking them in a tiny cage without interaction or anything to do will drive them crazy, but if they get their needs met, they should be ok if they aren't kept amused 24/7.
One thing my birds do LOVE is a hunk of tree from out the back (I have a eucalyptus in the back yard). Have you tried some natural vegetation with your fellow? (don't give eucalyptus to non-australian natives, but there'll be something your guy likes?)