I have been a Catholic all my 35 years. But I am becoming increasingly angry at how the Church abuses its power and, among many things, excludes Gay people from getting married. How can I come to terms with this?

You exemplify an interesting phenomenon wherein the modern world, which is often disdainful of Church “power,” then turns and expresses exaggerated notions of Church power.
 
In terms of divine moral law, the Church has no authority whatsoever to overturn the biblical teaching against homosexual acts, or to redefine the parameters of marriage as given by God in the Scriptures and Sacred Tradition. The Church is the servant of the Word of God (cf Catechism #86), not an all-powerful entity that is able to tear pages from the Bible, cross out lines, or overrule it. The sinfulness of homosexual acts (and also illicit heterosexual acts, such as fornication and adultery) is consistently taught at every stage of biblical revelation to the last books.
 
Hence, I would urge you to reconsider that what you call an abuse of power, is actually, a humble recognition of the limits of her power by the Church.