Not necessarily. If we think of original sin as uncleanness as conceived in the Old Testament, it is something that can be reversed. This is more of an eastern view than western, but Mary’s sin would not have passed on to Jesus because the mere arrival of the Incarnate Word would have, as with the woman who touched the hem of Jesus’ garment, purified her womb, removing any uncleanness.
The idea of uncleanness from the Old Testament is the best model for sin; indeed that’s the only reason for the concept in the first place! And under that concept one can become unclean by choice but also by accident. “Original sin” can mean two things: the sin that originated sin among humans, and the sin we are born into – the first act of uncleanness that made all later humans unclean, and the uncleanness into which we are born. My sister the quality engineer use the example of manufacturing off of a prototype: if the prototype is damaged, and all further units are based on the prototype, then they are also all damaged. That fits well with the “fallen short” model, that there is a standard we are supposed to meet but we are born already short of that standard because our “prototype” fails to meet that standard. It also fits well with the preferred model in the east, where sin is regarded as spiritual disease that is passed on. Today we might say that sin is a spiritual genetic flaw that manifests itself in actual sins; some ancient theologians regarded sin as pollution of the (metaphysical) nature of humanity, a pollution that we all share because it is totally intermixed with human nature.
Indeed it’s only under such a model that “the Lamb of God… takes away the sin of the world” makes any sense: “sin” is singular and is an attribute of the whole world. In the west theologians fell into a legal model where sin was all about individual actions that transgressed specific commands, but that fails to take into account the idea of sin, singular, as belonging to the whole world. And that sin-as-nature is the only concept that really reflects the depth of separation from God expressed as referring to us as “strangers” and “aliens”, terms that Oswald Hoffman, the great Lutheran Hour preacher, illustrated as being born into the wrong camp: we are enemies by nature, according to Paul, automatically soldiers on the wrong side because of ‘where’ we were born.