I have the same problem with my daughter who needed blood drawn since she was about 6 years old. It seemed to get worse as she grew older and got the idea of it. Yes, they also used infant needles.
There is a cream that numbs the skin topically that could be applied half an hour before the needle goes in, it is prescribed by our doctor: EMLA Cream (lidocaine 2.5% and prilocaine 2.5%)
She may have done better with the cream on, but the last few times (I forgot if she had the cream or not), she was screaming so hard, 3 people had to handle her so she wouldn't move. It could be psychological also, I can imagine my being uncomfortable/ nervous before my blood drawn. I feel so bad for her that I had given up this diagnostic procedure once it was deemed not so necessary.