These are all great suggestions!
I live in an apartment, and Miles also has separation anxiety and used to bark nonstop when I would leave. He's calmed down a little bit over time (he's over a year old now), but still gets anxious every now and then.
You sound a lot like me in the way that I do NOT want to disturb my neighbors! Since changing the barking behavior might take a while, I suggest talking to your neighbors. It's amazing how much that can help the situation. Rather than waiting for them to approach you about the barking, go to them first. Kindly explain that you've got a young puppy with separation anxiety, and you are aware that she might be a bit noisy right now. Tell them you're definitely working with her and feel she'll get comfortable soon. Lastly, explain to them that they can call you if they have any questions or concerns. My neighbors appreciated this SO MUCH when I got Miles. I think the fact that I was respectful to them in the beginning made them a little more patient when they would hear Miles.
It was a huge load off my shoulders to not have to worry about whether or not my neighbors were upset with me.