Does the "sincere repentance" mentioned in the supplication mean abandoning the sin that a person has committed, or does it include complete righteousness and adherence to obedience in all aspects of life, and what are the implications of that for atoning for a vow?

Sincere repentance is that which fulfills its conditions: immediately abstaining from the sin, regretting it, and resolving not to return to it. Repentance is accepted with these conditions, even if the sin warrants a prescribed punishment (hadd) and it has not been carried out. A Muslim should conceal himself if he has committed an offense warranting a hadd punishment and promptly initiate sincere repentance. Righteousness is a sign of accepted repentance. If a servant returns to a sin after his initial repentance, this does not invalidate his previous repentance, and he must repent for the new sin.