Am I obligated to continue financially assisting my family, even after their situation has improved, as long as it makes them happy, especially since my wife objects to it and I am experiencing financial hardship?

Our advice to you is to continue to maintain ties with your family to the best of your ability, but do not neglect yourself and your children. Save for them and for any needs that may arise for you, for balancing these matters is more preferable and has a better outcome.