Do even natual-born Spanish speakers make grammatical/spelling mistakes?

Almost everyone makes mistakes in their own language (native tongue). You can't compare yourself to your cousins. Like you said, they speak Spanish everyday, and it is possibly the only language they can speak fluently. You should be proud for being bilingual. What I would suggest is that you read books in Spanish and watch programs on TV (like the news) to better yourself in the language. Maybe you could also look up information about your doubts in the language on the internet! Good luck!