I feel it is one of those AmE vs BrE differences. Just to make sure that I understand both of you correctly...
...anxious to share the good news with you (Clive - BrE - anxious and excitednot synonymous) ...anxious to share the good news with you (AStar- AmE - anxious and excited
They are related, but not synonymous. Anxious can have a sense of negativity, excited does not. Both have the idea of not being able to relax.
I am anxious to see the grades on the test. (I haven't seen them yet. I am worried that I might have done badly.) I am excited to see the grades on the test. (I got an "A".)
excited: very happy and enthusiastic because something g