Interesting discussion. I'm not in the US and I'm not a US citizen, so I can't answer the specific question. However I do tend to vote on the left (but not always). Here's a few of my thoughts, hopefully they will help answer the OP's question.
- I don't care about the motivations or "values" of politicians, all I care about is what their policies are, and what likely affects and outcomes these policies will have. We can never know their true motivations or values. They might decide to tell us, but even then they are probably lying.
- I then look at these policies and potential outcomes, and decide how they measure up with my understanding of God's principles.
- There will always be a balancing act. There will be policies on both sides that I am not happy with, and there will be policies on both sides that I am happy with. So I weigh them up, and then decide.
- When weighing up one has to put a "weight" on each issue/policy. Some things are clearly more important than others. This is extremely difficult and very personal. Even though you and I may have very similar beliefs, I don't expect you to assign the same weights as I do. The weighting is done carefully and prayerfully, and I will justify them to no one except God.
OK, as an example I'll attempt to address some of the issues:
Abortion: The issue here is not who "supports" abortion and who doesn't. The issue is will abortions occur, and what will be the result of these abortions? If abortions are outright banned, they will still occur. But they will occur in secret. Those who are wealthy enough will pay too much to a corrupt but skilled medical professional. The rest will get botched procedures resulting in unnecessary suffering and death. So although I am opposed to abortion, I will not support a party who proposes an outright ban on abortion. I will also not be keen to support a party who proposes abortion-on-demand, no-questions-asked. I am keen to minimise abortion, but I am just as keen to ensure that any abortions that occur are as safe for the woman as possible.
Trans / Gay: I don't believe that anyone should be discriminated against. So I will not support any party who tries to take rights away from anyone, even if I don't agree with their lifestyle. I don't include SSM in this. Not because I don't think it's a rights issue, it is. But because it's way too complex for this discussion. In short I would not support SSM, but the reasons are complex and irrelevant to this discussion.
Child out of wedlock: I don't believe that anyone with a child out of wedlock should be disadvantaged. In fact I would say that we should support them if the father is not around to help. Women are already disadvantaged in our society, discriminating against single mothers will not help. You can't legislate morality, especially if it's a morality that not everyone shares.
Foreign policy: The US supported by her allies have been the aggressors in too many wars, often for the wrong reasons. I will tend to support those parties who support more peaceful methods of diplomacy.
Economic justice: This is a moral issue just as much as the others. The USA has a middle-class which is shrinking. The rich are getting richer and the poor are getting poorer. I visited your great country last month (WA) and I was shocked by the poverty I saw. Now maybe poverty is unavoidable, but even so we should do all we can to reduce poverty and its effects. This issue rates very important when measured against God's principles. I will vote with this in mind. Universal healthcare and minimum wage are moral issues. In my assessment they are more important than, for example, gay moral issues. So when deciding my vote I would place more weight on reducing poverty (as a moral issue) than I would opposing a gay (moral) issues that I disagree with. Unfortunately there will always be a compromise, unless one refuses to vote at all.
Stewardship: Pollution and climate change. God has given us this planet and all that is in it for us to care for. He expects us to look after it, and we are not doing a good job. So I am extremely reluctant to offer my support to a party who doesn't strongly support ways to reduce carbon and other pollutants in the atmosphere. This one is a double whammy because it's not just a stewardship issue. The effects of climate change will impact the poorest first and more severely. God has a heart for the poor, and He expects us to be the same.
It's these last two issues that tend to push me towards voting left. The Republicans and their equvalents across the globe tend to rate very poor on these two. Make no mistake these issues are moral issues close to the heart of God, just as much as abortion and LGBTI are. No political party is a good fit for Christians. There is no "correct" party for Christians to vote for. We just have to assess the policies of each, and vote for which one measures up to God's standards the best. It's hard work, but I believe that God expects us to use our brains and prayerfully decide.
One last thought for now: Secularism is a good thing for Christianity. Without it you would not be free to explore and discover God. You would be told what to believe, and punished if you are not seen to agree. Even if what you were told to believe were true, there would be no joy or conviction in it. Secularism protects us as believers. Yes it also protects false religions, but that is the price we pay for freedom. I'm pretty sure God wants us to work out our salvation for ourselves. Secularism is not an emptiness, it is a separation of responsibilities. It's no more an emptiness than different rolls for wife and husband is an emptiness.