How is one to be delivered from sins related to desires, and how can one ensure the supplication for a righteous marriage is answered after repeatedly falling into sin and losing hope?

The questioner is correct in her judgment that unrestricted mixing (ikhtilat) devoid of Shariah-compliant controls is a sin, especially with a maternal cousin, due to the prevalent leniency. One must immediately cease the sin, feel remorse, and resolve not to return to it. Despairing of Allah's mercy is greater than any sin, so the fear of a sin should not lead one into a greater sin. It is not permissible to abandon supplication (du'a) due to impatience for a response.