Let's sort out the questions:
Is there any culpability for addictive sins?
If the initial actions that led to the addiction were sins, yes, there is culpability for the actions that were undertaken with the appropriate knowledge and consent. If those initial sins have since been confessed and forgiven, there is no longer culpability for them, but there may remain temporal consequences (which can include addiction).
Does addiction diminish culpability for "addictive sins"?
Insofar as addiction impairs one's ability to consent to the sin, yes, addiction may lessen one's personal culpability for a later sin committed under the influence of the addiction. From the Catechism of the Catholic Church
One commits venial sin when, in a less serious matter, he does not observe the standard prescribed by the moral law, or when he disobeys the moral law in a grave matter, but without full knowledge or without complete consent (CCC 1862).
This does not mean that there is absolutely no culpability for addictive sins; only that addiction may impair consent and diminish culpability. In such a case, depending on the capacity for consent, an addictive sin may not be a mortal sin if it lacks full and free consent. A sin committed with insufficient consent would be venial.
Should I confess addictive sins?
Even if one's otherwise grave action is not a mortal sin because of diminished consent, it can still be profitable to confess sins that are committed because of an addiction. Confessing the sins and obtaining the grace of absolution can strengthen the person in his or her fight against the addiction. The Church encourages people to confess venial sins:
Without being strictly necessary, confession of everyday faults (venial sins) is nevertheless strongly recommended by the Church. Indeed the regular confession of our venial sins helps us form our conscience, fight against evil tendencies, let ourselves be healed by Christ, and progress in the life of the Spirit. By receiving more frequently through this sacrament the gift of the Father's mercy, we are spurred to be merciful as he is merciful:
Whoever confesses his sins ... is already working with God. God indicts your sins; if you also indict them, you are joined with God. Man and sinner are, so to speak, two realities: when you hear "man" -- this is what God has made; when you hear "sinner" -- this is what man himself has made. Destroy what you have made, so that God may save what he has made.... When you begin to abhor what you have made, it is then that your good works are beginning, since you are accusing yourself of your evil works. The beginning of good works is the confession of evil works. You do the truth and come to the light (CCC 1458).
The spiritual effects of the sacrament of penance are:
- reconciliation with God by which the penitent recovers grace;
- reconciliation with the Church;
- remission of the eternal punishment incurred by mortal sins;
- remission, at least in part, of temporal punishments resulting from sin;
- peace and serenity of conscience, and spiritual consolation;
- an increase of spiritual strength for the Christian battle (CCC 1496).