At this point in our history, the record of church groups in lobbying state and federal governments on these issues is not very strong. The churches have, for the most part, not gotten what they want in these areas. There are laws that they have managed to get passed (particularly in the area of abortion), but they have not achieved their main goals in any of these areas.
Prohibition is not really on the agenda of most churches these days. I do not believe that there is a strong lobbying effort for going “dry” in almost any part of America today. Thus, I would say that churches have not been at all successful on this issue (to the extent that they even care) in a very long time.
The record is only slightly better on the issue of birth control. There are many churches that do not condone the use of any birth control at all. To the extent that they push for bans on the use of birth control, they have not been successful. There have been some minor wins for such churches. For example, it is somewhat difficult to get the “Plan B” pill in many states even though the FDA had found that the drug should be available over the counter. However, this sort of victory is relatively minor when compared to the easy availability of most contraceptives.
The idea of restoring prayer to public schools has also largely failed. This is largely because any law that allowed overt and mandatory prayer in schools would run afoul of the Supreme Court, which is very difficult to lobby. Therefore, there are no states in which public schools can legally hold official prayers mandated by school authorities. You could say that churches have had some success in getting student-led prayers to be allowed in school. However, this is very far from the sort of prayer in school that was prevalent in the mid-20th century.
Finally, we turn to abortion where the churches have had the greatest success. Here, too, they have not won their greatest desires. Roe v. Wade is still the law of the land and that means that abortions cannot be banned. However, churches and other groups have succeeded in making it harder and harder to get abortions in many states. They have, for example, gotten many states and the federal government to ban the use of government money to provide abortions. They have made it harder for abortion clinics to be licensed. These are real victories, even if they fall short of outlawing abortions.
Thus, we can see that on most of these issues, churches have not been very successful in lobbying the government. The same thing seems to be happening with respect to the issue of gay marriage. It would appear that church efforts are, at this point in our history, not very effective in achieving their policy goals.