My girl also gets over excited and bites; she doesn't mean to but she just does.
Budgies, especially those we have a very strong bond with, can read our body language just as we read theirs. When my girl bites, I turn to her with a hurt face and say "no, Mallorn, gently."
She has known for a while that "No" means "No", but over the years she has also learned that "gently" means that she's hurt me and therefore needs to lighten up on the beak. When I offer her my hand afterwards, if she is gentler with me I praise her heavily and smile at her. It seems to work as she rarely bites me and when she does, she knows she did the wrong thing. Maybe something like this would work?