First of all she is not technically your cousin. She would be your third cousin if not mistaken. You do your research on this one to confirm i'm not even sure
Is it illegal? In most states it's only illegal if it's your cousin (daughter of your direct aunt/uncle). Any cousins past that it's ok to even marry in most states. You two are pretty far removed i'm sure you could get married in all 50 states so there's nothing illegal about it.
Is it morally wrong to have secks with cousins?: It's allowed (and encouraged) in the Bible and that was for actual cousins. In your case you are far more removed genetically than actual cousins since you would be a third cousin or whatever. The only forbidden secks in the Bible is homosexual sex/siblings/parents/bestiality/secks with the wife of your brother/another man's wife (anything I miss?).
Secks with cousins is not forbidden but was encouraged by God in a few cases. So if you go by the Bible for morals there's nothing wrong with cousin secks no matter what anyone tells you.
10% of the world's population are cousin marriages. It's a fairly common thing everywhere except here I guess. It was also common here in the west till the laws were changed not too long ago from what I understand. Marriages between cousins tend to last. I guess it's due to the actual family connection. You already have sort of a bond with them.
Personally I see nothing wrong with it. Secks with an actual cousin (daughter of direct aunt or uncle) would be a little too close genetically in my opinion but anything after that I see nothing wrong with it.