I would first like to thank you for asking in a polite and constructive way! The following answer is based on my experience as a member of a congregation which is accepting of QUILTBAG people - a category which includes the minister - and their loving relationships; not everyone is in favour of same-sex marriage, but we've talked about it quite a bit. I have affection and respect for people who disagree, and I hope that we can all get along as we learn about this issue in good faith.
There are two sides to the answer. You asked about one of them, which is how to deal with the apparently anti-gay passages in Scripture. The other one, which I'll get to as well because it is closely related, is the positive case for affirming homosexual people and relationships on the same level as heterosexual ones. (The title of your question asks about marriage, and to get all the way over there one must do a bit more than just say that homosexual behaviour or activity isn't that bad.)
As far as the "clobber verses" go, the usual answer is to distinguish between what it is that homosexual people do today, and what is being condemned in those passages. Today, the idea of "same-sex marriage" is proposed to be exactly the same as opposite-sex marriage, except that the two people in the marriage are the same sex. In all other respects - love, mutual trust, shared commitment, and so on - it is meant to be identical. But this is indeed a new idea, and is not characteristic of the way that relationships between people of the same sex were construed in the ancient world. In a system where the institution of the family hinges on the dominant presence of the paterfamilias, typical male homosexual relationships ended up being socially marginal (no inheritance is going on here) and unequal (the penetrator being treated differently from the penetrated - the latter was shameful because he had degraded himself to the level of a woman). The point is that what's being condemned may be the unequal, abusive, lust-filled sort of relationship that would have come to Paul's original readers' minds when they thought about men in sexual relationships with other men.
Accompanying textual criticism has tried to understand just what Paul might have meant by his neologism arsenokoites, for example. This is the Greek word that is often translated as "homosexuals" or "sodomites" in the two NT passages you quote. It seems that there's definitely a sense of men (arsenes) going to bed/having sex (koite) but some have argued that he was upset about paedophiles, or procuring male prostitutes, or something else, in addition to the option I presented in the paragraph above. Lesbians do not seem to be treated here.
There is also a strand of interpretation that says that Paul did mean all homosexual people, but that he was wrong - and equally, that the Law of Moses did intend to condemn everyone who participated in a male homosexual act to death, but that it was wrong to do so. This differs from what I've mentioned so far because it doesn't even attempt to recover anything from Paul's words, instead only dismissing them as the product of a flawed culture. (In so doing, some obvious other problems are created.) This view is probably quite common but it is far from universal.
Now at last, the positive case is about believing that the love that exists between a couple is the same love regardless of their sex, and that their love is of divine origin. In other words,
God, who knows the human heart, testified to them by giving them the Holy Spirit, just as he did to us; and in cleansing their hearts by faith he has made no distinction between them and us. Acts 15:8-9 (NRSV)
Peter was speaking there about whether Gentiles could become Christians, but he is also making a general point about the testimony of the Spirit. We Gentiles can be saved because
There is no longer Jew or Greek, there is no longer slave or free, there is no longer male and female; for all of you are one in Christ Jesus. Galatians 3:28 (NRSV)
Recall that for Paul, these three distinctions were all about social caste. There is still such a thing as male and female identity, but what has vanished is the associated power relationship and privilege that denies proper love. This goes right back to the divergent ideas about homosexuality I spoke about above. Christians who support same-sex marriage find that it fits better with passages like these than with the clobber verses. Moreover, the ideas expressed here by Peter and Paul are felt to be closer to the heart of the Gospel. While this is very far from an open-and-shut case - there's a ton of stuff to consider about natural law, reproduction, and so on - it is the central pillar of the argument.