Is it permissible for a married woman to continue publishing da‘wah messages online, and should she answer a young man who asks about her marital status, for fear of causing fitnah in people's hearts?

The internet is a useful tool for da'wah (inviting to Islam) and spreading good, but it can also be a cause for evil. It is not permissible for you to respond to the young man, as this question could be a fitnah (temptation/trial), and blocking the means [to evil] is obligatory. Allah Almighty says: "O you who have believed, do not follow the footsteps of Satan. And whoever follows the footsteps of Satan - indeed, he enjoins immorality and wrongdoing" (An-Nur: 21). It is better for a woman to invite women of her own gender through women's forums, and to leave the da'wah of men to men, as this is better and safer in outcome.