How can one deal with a father who does not work and is indifferent to the suffering of his sick wife, who supports the household, while avoiding him due to his behavior and lack of being a role model?

If your father is abusive and negligent, his right to dutifulness and companionship with kindness does not diminish. Therefore, it is not permissible for you to abandon him or treat him badly. Rather, you must advise him gently and courteously, for honoring one's parents is among the best deeds and most beloved to Allah.