So not sacramentally married = not married at all?
No.
Two baptized Christians have a sacramental marriage. A marriage between non-baptized people or between one baptized and unbaptized person is a natural marriage. Natural marriages are valid.
However, Catholics (and only Catholics) are required to abide by Catholic canon law and get married in the Church or get the appropriate dispensations to be married elsewhere (and to a non-Catholic or non-Christian). The OP did not do that, and so his marriage (from a Catholic point of view) does not exist. If his wife were to agree, he could do what’s called a “convalidation,” and all that means is a Catholic wedding. Typically, convalidations are done without all the bells and whistles of a regular marriage, but it’s the same ceremony and the same vows.
ETA: To clarify, if the OP was Protestant, his marriage would be presumed valid *and *sacramental because both he and his wife are baptized. If he had later decided to become Catholic, it may have caused similar issues to the ones he’s facing, but the presumption would be that his marriage would be valid. It would take an investigation by a marriage tribunal to see if there was some other flaw that would have prevented the marriage from being validly contracted (there’s lots of grounds, but say, for instance, a defect of consent - one person by reason of mental illness could not make the promises of marriage). I’m not a canon lawyer, so I don’t have the whole measure of it.
But since the OP is Catholic, and didn’t marry in the Church, his marriage does not enjoy the presumption of validity, and due to the severity of the issues he’s facing it might not be wise of him to make it so. And his wife has no desire to, either, which means until/unless she does he will be persisting in grave sin, which is a dangerous place for a soul to be. Already he’s so dejected he’s quit practicing and is now committing even more grave sins. Catholics believe that spouses should help each other become holier to help us get to heaven. That is not what’s happening here, not one bit.