No the Church of Christ doesn't allow abortion, gay marriage, etc....That is the UNITED Church of Christ you are speaking of. There is a difference. I am a christian and am a member of the Church of Christ. It is a conservative group that follows the bible and Jesus' teachings and examples. Unfortunately people confuse the two because of the name.
I know there are some conservative churches of christ, but they all come from that same reconstruction stuff from the 1800s. The disciples of christ come from the same time, also, if my memory serves correctly.
Churches that follow God's Annointed One (Messiah/Christ) will be true to scripture. They will have a strong regard for Christians of past eras. They will teach the true good news that "Christ died for our sins according to the scriptures."
Those called out ones of God's Messiah will not be defeated and need "reconstructing." "The gates of hell shall not prevail against them."