Is not keeping ties with one's cousins considered a severance of the ties of kinship and a sin upon the one who does it, and what is the way to rectify that?

Your paternal cousins have three rights upon you: Islam, kinship, and邻里 (neighborliness), and you owe them the same. Injustice, abuse, or harm to body or honor are not permissible. Severing ties of kinship and abandoning them without a valid reason is a door to sin, and reconciliation and forgiveness are better. Extend your hand for reconciliation, be gentle with them, be patient with harm, and repel evil with good, for Allah Almighty says: "Repel [evil] with that which is better; then behold, he between whom and you was enmity [will become] as though he was a devoted friend." However, if maintaining ties with them causes harm, then there is no sin in keeping away from them, avoiding mingling with them, and merely exchanging greetings upon meeting. This is permissible 'beautiful avoidance' (hajr jameel), for Ibn Abd al-Barr said: "They (scholars) have agreed that it is permissible to avoid someone for more than three days if speaking to them brings a deficiency to the speaker in his religion, or causes harm to him in his person or his worldly affairs. For perhaps beautiful avoidance is better than harmful mingling." Beautiful avoidance is to abstain from mingling without causing harm.