There are roughly 4 answers given by different churches down the ages. The two fringes are the Judaisers, who argue that you should adopt all the law of Moses and got slapped down at the Council of Jerusalem in Acts 15, and the anti-nomians who argue that you don't have to obey any moral law at all - though they get challenged by Paul's comments in I Cor 6

9Do you not know that the wicked will not inherit the kingdom of God? Do not be deceived: Neither the sexually immoral nor idolaters nor adulterers nor male prostitutes nor homosexual offenders 10nor thieves nor the greedy nor drunkards nor slanderers nor swindlers will inherit the kingdom of God. 11And that is what some of you were. But you were washed, you were sanctified, you were justified in the name of the Lord Jesus Christ and by the Spirit of our God.

12"Everything is permissible for me" - but not everything is beneficial. "Everything is permissible for me" - but I will not be mastered by anything. 13"Food for the stomach and the stomach for food" - but God will destroy them both. The body is not meant for sexual immorality, but for the Lord, and the Lord for the body. 14By his power God raised the Lord from the dead, and he will raise us also. 15Do you not know that your bodies are members of Christ himself? Shall I then take the members of Christ and unite them with a prostitute? Never! 16Do you not know that he who unites himself with a prostitute is one with her in body? For it is said, "The two will become one flesh."

The two main stream responses are

1) to point to the 'moral law' within Moses and see that as still being applicable whilst the ceremonial and food laws are to be ignored

2) to take a more radical approach seeing the presence of the Spirit as guiding the believer - but knowing that guidance is only valid if it conforms to certain norms. This is remarkably difficult to sustain - the propensity to fall either towards legalism or licence instead of depending on God is always there - with the result that churches tend to end up with their own laws, the most frequent of which is of course 'no alcohol'...

Which is why the proof texts on the gay issue from the Old Testament shine relatively little light on the debate, although there is an interesting recent argument that the Greek words used by Paul in his I Cor 6 condemnation of gay behaviour actually derive from the Hebrew. :)

Don't know if that helps any!