It seems to me that there are some false premises in the OP’s concerns that she raises.
The first one is that morality is confined to religious creeds (institutionally or privately), and that thus civil society is by definition amoral (or should be). If this were true, only believers could claim a moral orientation, but we know that is not true. There are plenty of atheists and agnostics who make moral arguments on public issues. Similarly, politicians, and journalists writing about issues, concern themselves often with what is explicitly “moral.” They refer to that on the House and Senate floor - and I’m not referring to Life issues here, but issues such as jobs, war, and more. Congressional hearings are especially surrounded with moral language, such as in discussions of corporate responsibility toward individuals and society. Do you think that when a newspaper columnist decries Wall Street deception or Real Estate foreclosure practices as “immoral,” he or she is speaking from a religious point of view? The entire recent economic/governmental crisis in Wisconsin was framed by many secular commentators as a moral crisis in addition to a practical one. Accessibility to adequate education is most often characterized by secularists as a moral issue. Is that enough? (Not being sarcastic, just don’t want to belabor what should be obvious to all of us, I think.)
It is actually a fairly contemporary development, relatively speaking, that morality has been shifted & confined to religious institutions. It began in earnest about 35 years ago, and became a much more aggressive effort during the '80’s and '90s. And that has happened more from the Left, less from the Right, although frankly both the Right and the Left do it: The Right sometimes to evade corporate responsibility, the Left often to evade private responsibility. But in any case, this being an era which pursues unlimited guilt-free pleasure, naturally those most interested in protecting that pursuit are also the ones most interested in silencing the voices of morality (because moral constructs imply limits).
This brings me to the second false premise: that our society permits, would permit, the “forcing” of a specifically religious (sectarian-identified, such as Catholic, Mormon, Jewish) morality. Advocating is not forcing. We have freedom of religion in this country, not freedom from religion. It means that religions can express a viewpoint – yes, a moral viewpoint – in the public square, about public issues which affect everyone or merely someones in society. They get to do that, despite the annoyance it may cause anyone. We will never be a theocracy; we certainly are not in danger of becoming one in the near future. We are, however, increasingly in danger of the dictatorship of relativism (which eviscerates morality, being that the definitions are forever shifting and subjective). If anything is being “forced” on society – and has been for at least the last 20 years – it is immorality. (What must be in public school curricula, what explicit sex scenes and vulgarity exists unavoidably in the popular media, etc.)
So both an immorality and an amorality are being artificially imposed on a nation whose founding ideals implicitly embraced the morality of the common good, and whose institutions have always, except quite recently, embraced the development of character, and sought people of character to represent them (schools, governmental bodies, & more).
Rather than “forcing” morality on others, the Catholic Church is among a handful of bodies that remains a refuge for those seeking a morally coherent & consistent code. Like other religions, it has a legitimately moral role to play in preserving a healthy direction for society. It encourages the adoption of perimeters, without which a society can dissolve from within. (Even a non-religious moral system supports the function of order.)
Finally, the third false premise – if not by the OP, certainly adopted by contemporary society – is that morality refers specifically or mainly to sexual mores. Morality encompasses an awful lot, from the viewpoint of believers and nonbelievers alike. Civil disobedience, for example, is an outgrowth of many moral causes. Liberation movements internationally, and governmental policies about migration – these are described in moral terms by churches, by individuals, and by governments.
Peace 
