Is there a way a man and woman can get a civil union or domestic partnership...

A civil union or a domestic partnership does not provide a couple with the same rights and obligations as being officially married. They can enter into an agreement, which might be enforceable as a contract in a non-family law court, but it is not part of what a family law court does.
In Florida, there is no such thing as common law marriage. If you're living together, even with an agreement, you are not married, and you are not recognized as being married, and are not afforded the laws, the rights and obligations that are afforded to officially-married individuals.