It seems so far like the staff are tolerating topics on this in various sections, probably because it is such a water-shed moment. So I guess I will give my take for now.
Under the theocracy of Israel God imposed the death penalty for various things, including homosexuality, but also for such things as breaking the Sabbath.
We would not expect a modern secular government to do such. Paul, writing during the time of the emperor simply speaks of submitting to the government. As Americans we have some limited role in the government. To the degree that we have a role we should try to support the role that God assigns to government:
Rom 13:4 For he is the minister of God to thee for good. But if thou do that which is evil, be afraid; for he beareth not the sword in vain: for he is the minister of God, a revenger to execute wrath upon him that doeth evil.
Rom 13:5 Wherefore ye must needs be subject, not only for wrath, but also for conscience sake.
Rom 13:6 For for this cause pay ye tribute also: for they are God's ministers, attending continually upon this very thing.
The role of the government is to uphold good and punish evil.
However, this is not a theocracy. While Christians may advocate for their view as anyone else can, the government still operates within various confines that are beyond control without a constitutional amendment, etc. which it does not look very likely. There is not support for it among the populace. From a secular society view there is no compelling reason to deny rights to these people.
Ultimately as Christians we may have to simply return to the advice that Paul gave while under an emperor. We do not have to agree with everything the government does. But we do have to submit when conscience allows.
As for judging, we are to judge those in the church, not those out of the church. Why would we expect those outside the church to live as we would?
1Co 5:9 I wrote unto you in an epistle not to company with fornicators:
1Co 5:10 Yet not altogether with the fornicators of this world, or with the covetous, or extortioners, or with idolaters; for then must ye needs go out of the world.
1Co 5:11 But now I have written unto you not to keep company, if any man that is called a brother be a fornicator, or covetous, or an idolater, or a railer, or a drunkard, or an extortioner; with such an one no not to eat.
1Co 5:12 For what have I to do to judge them also that are without? do not ye judge them that are within?
1Co 5:13 But them that are without God judgeth. Therefore put away from among yourselves that wicked person.
Within the church we are to judge, not only homosexuality, but liars, continual gossipers, those who are continually drunk, fornicators, adulterers, etc.
However, even within the church the primary purpose of this is to be redemptive.
Co 7:8 For even if I made you grieve with my letter, I do not regret it—though I did regret it, for I see that that letter grieved you, though only for a while.
2Co 7:9 As it is, I rejoice, not because you were grieved, but because you were grieved into repenting. For you felt a godly grief, so that you suffered no loss through us.
2Co 7:10 For godly grief produces a repentance that leads to salvation without regret, whereas worldly grief produces death.
We are to try to restore:
Gal 6:1 Brothers, if anyone is caught in any transgression, you who are spiritual should restore him in a spirit of gentleness. Keep watch on yourself, lest you too be tempted.
Gal 6:2 Bear one another's burdens, and so fulfill the law of Christ.
As to public speech, we should stop, as Christians, advocating for our "rights". The early church did not win over the world by claiming their rights, but by suffering unjustly while loving those who spitefully used them.
We cannot hold back the gospel or preaching what Jesus has said.
Act 4:18 So they called them and charged them not to speak or teach at all in the name of Jesus.
Act 4:19 But Peter and John answered them, "Whether it is right in the sight of God to listen to you rather than to God, you must judge,
Act 4:20 for we cannot but speak of what we have seen and heard."
Speaking to a church in a hostile environment Peter said:
1Pe 2:12 Keep your conduct among the Gentiles honorable, so that when they speak against you as evildoers, they may see your good deeds and glorify God on the day of visitation.
1Pe 2:13 Be subject for the Lord's sake to every human institution, whether it be to the emperor as supreme,
1Pe 2:14 or to governors as sent by him to punish those who do evil and to praise those who do good.
1Pe 2:15 For this is the will of God, that by doing good you should put to silence the ignorance of foolish people.
1Pe 2:16 Live as people who are free, not using your freedom as a cover-up for evil, but living as servants of God.
1Pe 2:17 Honor everyone. Love the brotherhood. Fear God. Honor the emperor.
1Pe 2:18 Servants, be subject to your masters with all respect, not only to the good and gentle but also to the unjust.
1Pe 2:19 For this is a gracious thing, when, mindful of God, one endures sorrows while suffering unjustly.
1Pe 2:20 For what credit is it if, when you sin and are beaten for it, you endure? But if when you do good and suffer for it you endure, this is a gracious thing in the sight of God.
1Pe 2:21 For to this you have been called, because Christ also suffered for you, leaving you an example, so that you might follow in his steps.
1Pe 2:22 He committed no sin, neither was deceit found in his mouth.
1Pe 2:23 When he was reviled, he did not revile in return; when he suffered, he did not threaten, but continued entrusting himself to him who judges justly.
1Pe 2:24 He himself bore our sins in his body on the tree, that we might die to sin and live to righteousness. By his wounds you have been healed.
1Pe 3:13 Now who is there to harm you if you are zealous for what is good?
1Pe 3:14 But even if you should suffer for righteousness' sake, you will be blessed. Have no fear of them, nor be troubled,
1Pe 3:15 but in your hearts honor Christ the Lord as holy, always being prepared to make a defense to anyone who asks you for a reason for the hope that is in you; yet do it with gentleness and respect,
1Pe 3:16 having a good conscience, so that, when you are slandered, those who revile your good behavior in Christ may be put to shame.
1Pe 3:17 For it is better to suffer for doing good, if that should be God's will, than for doing evil.
1Pe 3:18 For Christ also suffered once for sins, the righteous for the unrighteous, that he might bring us to God, being put to death in the flesh but made alive in the spirit,