I always pray first! I've never stuck someone without saying "Lord, please guide my needle". I feel like it really helps me calm down and know that Gods in control and with Him, I can be a vessel for His greatness. Yes, even through an IV stick.
I dunno how spiritual you are, just wanted to share my coping technique. I feel it benefits me and my patient!!
Your confidence will build with time and success. I promise! Don't be so hard on yourself. Take some deep breaths before you go in and tell yourself "I can do this". If it helps any at all, try to spark a conversation with your patient while your in the room. It may help lighten then mood and get you more comfortable with your environment. Don't give up! Keep working and keep praying. You will get there!
Shake your hands and take some nice deep steady breaths, try not to focus on others if you're being observed. Deep down you know how to do the skill so just have confidence in yourself while maintaining patient safety and privacy. Good luck.