I don't really know what it's like to be in your situation. But I would say this because it worked for me: fake it till you make it.

TL;DR: Fake your confidence, make some friends that are cool but not overconfident, be the best of you by develop the best part in you and for you and those who appreciate you for who you are only.

When I started out in a new school and I'm an introvert too, I don't really know anybody. I was assigned to multiple groups for different subjects with complete strangers. I was nervous at first but I faked being not nervous around them. It takes a lot for me to do that but I did it anyway. I listened to their conversation, trying to figure out their personalities, what they like and whatever. Anyway I just try my best to blend into their conversation by making memes and jokes or give some constructive thoughts when the conversation shifts towards school projects or serious subjects. It might take a long time to do that but everything will work out in the end. And remember, timing and situation are very important, don't just make some random joke in random situation.

You said people just don't like you, perhaps it's because they don't know you just yet. Find the best part of you, think about what would make that part of you interesting to them. Try bringing up subject slowly, don't randomly bring up a subject that is irrelevant to the conversation. Don't rush, you have got time until you don't, we all are, including your cousin. So don't dwell on the present too much, move on when things fail and don't make the same mistake again.

Try to be a confident, don't be vulnerable in front of people especially your cousin. Unless it's the one you really trust, you can try and be a little vulnerable around them, but remember don't trust people too easily.

If your cousin doesn't care about you after you've tried your best, don't mind them, they're just jerks if they keep ignoring you because of who you are, just stick with your friends if you have. When you meet them in the hall just walk past them, back straight, looking forward (however don't look like those girls in movies that are just overconfident). That's what worked for me when my ex best friends started to ignore me when we went to the same uni.

Everything works out in the end. Don't dwell on present too much. To me, self development is the meaning of my life. To become a better person is what made me kind of get out of my existential crisis that lasted almost 12 years. If you ever have a feeling of "Fuck this, this is who I am, this is what I am, there's no point in everything", just breathe, and tell yourself that what you are now may not be what you are in the future. Try to be a better person for you and yourself only.

P.S. 1: Also if you really want to fake your confidence, don't do it too much, otherwise you might end up becoming your cousin.

P.S. 2: You asked should you ask them why. In my opinion, the answer is no, at least for now. However, if you really want their recognition after they call you lame, go ahead, but the answer might not be what you want to hear.

Best of luck to you, hope you will made it in the end.