Have you tried finely chopping the veggies into tiny bits and placing them on a bowl?

Since your budgie is so far scared and reluctant to try fresh foods and given the fact she is currently moulting, you can get her ready-made dry egg food.
You can find this at any specialized pet store.

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RIP sweet Tito (Summer 2008 - January 17th 2013).
You are missed and never will be forgotten.