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You need to get the license in the state where the marriage will take place, in this case, New Hampshire.
Yes, interracial marriage is allowed and legally recognized in New Hampshire. The state has no restrictions on marriage based on race or ethnicity.
Yes, the state democratic party platform for New Hampshire does include marriage equality.
Marriage between first cousins is prohibited in the state of New Hampshire.
This is no longer necessary as same-sex marriage is now legal in New Hampshire.