These are the only states that allow couples to create a common law marriage:
Yes, anyone can do a name change. In theory, most states allow you to legally change your name by usage only -- meaning that you simply start using your new name without any court action, and at no cost.
However, practically speaking, because you don't have a marriage certificate, you will need an official court order changing your name before you'll get government agencies and many private companies, such as banks and title companies, to accept your new name.
To learn more about getting your name legall changed, check out Nolo's Name Change section. For more information on how unmarried couples can document their relationship, see Living Together: A Legal Guide for Unmarried Couples, by Ralph Warner, Toni Ihara, and Frederick Hertz (Nolo).