Is the questioner considered the lawful wife of the man with whom she cohabited and with whom she signed an un-official marriage document, or of the man with whom her marriage contract was subsequently solemnized, especially given that one of the witnesses denies his signature on the un-official contract?

The customary marriage mentioned is Islamically invalid; it is outright fornication because it lacks the conditions of a valid Islamic marriage, namely a guardian, witnesses, and public announcement.

You both must offer sincere repentance and stay away from this man.

You are still the wife of your paternal cousin by a valid Islamic contract.

You should conceal what happened, as it was a result of violating God's law, which prohibits khalwa (definition) (private seclusion of a man and a non-mahram (definition) woman), commands lowering the gaze, and forbids everything that leads to fornication.