Can i get joint custody if my sons mom got a dui?

This is probably best posted to a family lawyer. If you used a lawyer to establish paternity and begin child support and you're truly interested in what's best for your son, contact that lawyer to see how your local judge will view your girlfriend's conduct. Usually one DUI is not enough to change custody, but if you've been involved in your son's daily activities, school, religious training and sports or other extra-curriculars, you should ask for joint custody.