I am really sorry to hear about your budgie as a budgie owner myself of one who isn't well, I know what you must be going through. Unfortunately I can't help with the knowledge your require, the droppings problem could be lots of things, infection, liver disease, bad reaction to something eaten.. etc etc. If she is on antibiotics you should give them a chance to work, but look them up on the internet for side effects... I find it informative.
As for the beak movement, is something stuck in her throat, does she have any blockage there or is it just that she has started this under stress? As I say, I'm not much help really but I wanted to answer because I know how desperate you can feel with no one to turn to over a sick pet.
Is it possible you can phone your vet under emergency just to ask for advice at least? I do hope you get the help you need and that your poor little birdie will soon recover fully and give you great pleasure again.
with very best wishes
Chrissie