There is nothing illegal about an unmarried couple living together. The couple generally can live wherever they wish. Some local zoning laws prohibit more than three unrelated persons from living together in one house or apartment, but those laws would not apply to a two person household. The government’s attempt to limit a couple’s right to live together would probably be considered a violation of the couple’s right to free association under the First Amendment to the United States Constitution.
A few states still have laws on the books prohibiting fornication sexual relations between two persons who are not married but such laws are almost never enforced. Some states also have laws against sodomy, which, among other things, prohibit sexual relations between people of the same sex. Those laws are rarely enforced if the conduct is private, consensual, and between adults (although in 1986, the United States Supreme Court in a divided decision did uphold a Georgia law criminalizing private sexual relations between two men).