In this case, using 'shy' is better. Though it is possible to feel 'ashamed' when you make mistakes, that's normally a word you only use if it really bothers you, and you feel deeply bad about those mistakes, as if you should apologize for making them.
Saying "I'm a little shy because I make mistakes" is a bit more laid back.
You can also say "I'm a little embarrassed because I make mistakes."