Do these people believe in the Old Testament but conveniently ignore the story of Sodom and Gomorrah?
1)It's not the "old testament." It's the Hebrew Bible (or TaNa"KH).
2)No, they don't accept the truth of the Bible. They have accepted liberal Protestant "historical criticism" almost from their beginning.
3)The prohibition on homosexuality doesn't come from Genesis 19 but from Leviticus 18. Homosexual rape was only one part of the wickedness of Sodom.