Is the marital relationship considered unlawful if the wife discloses information about her marital problems to others in response to their questions, and does this necessitate divorce, or are there other solutions?

What your husband did by conditioning the prohibition of conjugal relations upon someone knowing about your problems is an issue about which scholars have differed. The preponderant opinion is that its ruling depends on the husband's intention. If he intended ẓihār (a pronouncement by which a husband likens his wife to an unmarriageable female relative, making her unlawful to him), or divorce, or an oath, then he is bound by what he intended. It must be taken into consideration whether your husband had a specific intention when he said this, or whether the reason for his condition has ceased to exist. You should not have told anyone about your family problems; you could have used tawriyah (an ambiguous statement that is technically true but misleading). It is better for both of you to ask a scholar or to consult a Sharia court so that the judge can ascertain your husband's intention.