If your living arrangemen were to have taken place before the council of Trent or in an area where the decrees of the council wwere to have been unable to have been promulgated then you would have been married.
I think the Novus Ordo is teaching that marriage occurs between 2 individuals, and that a priest is not required. I suppose once the 2 individuals commit to marriage then preform the marital act they are then married. If I am understanding things correctly.
You are not understanding things correctly.
In order for Catholics to be validly married, even in the Novus Ordo, they must do so in a church before a priest. The priest does not confer the Sacrament, rather the husband and wife confer it on each other, but the priest witnesses the marriage for the Church and blesses it.
Even though the priest is only a witness of the marriage, Canon Law (even the new Canon Law) is clear that if a Catholic marries outside of the Church, it is invalid; in other words, there is no marriage at all.