@Miss anonymous ,
I'll assume, since you never shared the timeline, that you hooked up with your cousin before you got involved with your current boyfriend. I am aware that this is probably naive of me, but I want to believe the best before I have to accept the worst.
If you can't deal with the problem yourself, perhaps you can visibly show this cousin of yours that you pick your boyfriend over him. Ignore your cousin, ignore his drinking, and hang out with your boyfriend as much as possible. If your cousin keeps hassling you, perhaps you should get your boyfriend involved, explain what happened and how your cousin won't leave you be. He probably won't be pleased, but if he loves you, he might tell your cousin off.