At the moment, the OP is in a valid, yet irregular marriage.
No marriage is ever invalid unless and until the marriage is judged to be invalid by the Church and every canonist will tell you that one can never presume a marriage to be invalid apart from a tribunal. Since a finding of invalidity has not been made, the presumption is that the marriage is valid, but irregularly contracted. The marriage is* irregular*, meaning that certain aspects of the wedding were not performed consistent with the methodology of the Church. A marriage surely occurred, the unanswered questions is did a sacramental marriage occur?
If both parties were baptized, the marriage is possibly sacramental, but because marriage, like all sacraments involve both the correct matter and the correct form, the sacramental nature of the marriage is in question. Radial Sanitation and Convalidation correct an underlying defect(s) of form and backdate this correction to the day of the original civil marriage. Defects of form are not trivial matters but they do not automatically make the marriage itself invalid, but irregular.
No one can presume the invalidity based on the lack of proper form (which is serious) as it is a finding of moral certainly that is reserved only to a Church tribunal that been granted access to all of the facts of the matter at hand.
But since the marriage was irregularly contracted and the Sacramental nature of the marriage is in question, the OP should seek the counsel of a priest know for his piety and fidelity to the Church and her teachings. Her options at the moment are limited but it seems to me that she may:
(1) Abstain from martial relations, live as brother and sister, go to Confession and Communion while this situation plays out;
(2) Or maintain martial relations, abstain from Communion while this situation plays out hoping that reconciliation is possible;
(3) Or, seek the radical sanitation or convalidation, the latter of which seems unlikely at the moment.
Cutting off martial relations with an unbelieving spouse or borderline believing spouse is not something that one should do apart from serious spiritual direction, as the marriage is presumed to be valid (as all marriages are) by the church. Because of the irregular manner that it was contracted, the sacramental nature of the marriage is questionable and the full communion of both parties in the Catholic Church is not possible at this moment because the form of the marriage contract matters much.
But at this moment, I am 100% certain that they are married, but, irregularly so.