With the writings of Saint Paul, both flesh and spirit are intimately tied together in human nature. Living according the “flesh”, cuts the body and spirit apart, killing the soul. The spirit is oriented to God; Saint Paul encourages us to live according to our Godly-oriented nature.
Christ came to pay the price for Adam’s disobedience, which was death. Having paid the price, despite having a blameless soul, he was able to rise from the dead. The infinite value of his sacrifice allows us all to rise as well.
I believe you are correct that I am primarily contrasting with most specifically Calvinist beliefs; my main exposure to “Baptists” were actually through conversations with a Non-Denominationalist, so her beliefs might not be quite representative of Baptists in general. Calvinists believe that only the “elect”, though chosen by God, will be saved. Catholics believe that we “elect” ourselves by living according to the spirit, and strengthening it with the graces of the sacraments empowered by Christ’s sacrifice on the Cross.
The difficulty for the Catholic faithful is knowing that we must live holy lives so that do not take God’s grace for granted and distance ourselves. We’re so close, we have further to fall. It can extraordinarily frustrating that those with no understanding can live apparently “according to the flesh”, give in to all temptation, and not be cut off from all grace. Bless them, for the know not what they do. They place themselves at such spiritual risk, and cause untold damage. But Christ warns us not to be jealous of the laborers who worked one hour, because salvation is his gift to give, and their salvation does not detract from ours.
We must trust that whatever God’s choice it will be just.