That's great! I know how you're feeling. I've had the same problem for every job that I've had.
Be sure to look for examples and tips online. This one for example has great letters that you could use as a guide and inspiration: https://bit.ly/2UUoHSJ
At the moment I'm working for a game studio and and what I did was first write a draft and go over it again making the sentences look more formal. After that I sent it to friends and they helped me out with things I didn't notice before.
Seems like your nerves are getting the better of you. Try to not express your feelings too much and keep it professional. Your CV and portfolio should help you out a lot too, so refer to your portfolio.
There are a few tips on this page as well but I'm not a fan of the example letters.
Try to sell yourself, don't boast and stay truthful. You can do this! You just have to overcome your fears.