Hi, love. Read through this and here's some major things that I saw.

"me: I keep telling you I'm not blaming you! Why can't you swallow the fact that you've said some things that you shouldn't have said?"

and then,

"me: You make me feel bad about myself. I wish I'd die. You tear me down. You never lift me up. You've shattered what little of a self esteem I had left."

Here you're clearly trying to put the blame on her. You're saying that she tears you down and makes you feel bad about yourself. That you don't have self esteem because she shattered it, which do you really think is the case? Be honest.

"I just want you to see the way you hurt me sometimes! Why do I always have to get hurt!? Nobody likes me! I can never have a good relationship with anyone! I've always been nice to people now I'm mean and cruel! I'm pathetic both ways! Nobody cares about how I cry myself to sleep. No one cares about all of the pain that I feel. No one wants to be around me or give me a hug."

And here you're guilt tripping her. Trying to get her to feel your pain so that she'll validate your feelings and comfort you, perhaps. So that she'll tell you she's wrong and you're right. But mostly that you want validation and comfort and security, I think, because you don't feel comfortable or secure with yourself and you want her to tell you you're not a terrible person.

And I also notice a lot of 'I' statements.

I know how terrible this all feels. Right now you're very emotional and wanting your feelings validated and I think it might be good for you to attempt a conversation from her if she's willing where you're not blaming her or making her feel guilty or anything, get a small bit of validation from her if possible, but don't push her, and then, again, I think you should try to distance yourself from her for a while and sort through your thoughts/try to validate yourself and heal, learn to enjoy your own company, etc.

Your cousin seems really upset and at a loss for what to do or say to help you. She seems to not think she can and thinks that she's said things that she thought was helpful, and she's hurt over your behavior.

No one can save you. You alone have to (and, I believe, can) save yourself.

I hope you feel better. I'm sorry this is happening, and I know it's not easy to get better but I think it's possible. I'm sorry you're in so much pain.