The justifications aren't "flimsy." They're rubbish - bovine excrement, if you know what I mean. And no, by and large, they don't believe Donald Trump is one of them.
It's not "state's rights," either - unless that's a dog whistle for that other thing - you know, the one that split the Baptists back in the 1950s-1960s. If so, that's probably part of it.
But mostly, it's the stupid "culture war." The evangelicals - and a lot of Catholics, who also voted for Trump - believe that a Trump administration will make abortion go away, and make the GLBT community go hide in the closet again, so the fundies can pretend they don't exist (or discriminate against them if they're not hidden). They're fighting a losing war - our younger generations have largely abandoned the lines of this war - but that doesn't mean they won't win some battles in the meantime.
They're wrong, of course. Trump may not even pursue the culture war stuff - but even if he does, the Republicans can't make abortion go away. They can "outlaw" it, but stopping mifepristone imports won't be any easier than stopping marijuana or cocaine imports - and we know how that has gone. Similarly, they can't make the gay community go back "in" - and I somehow don't think that John Roberts wants to preside over a Supreme Court that issues rulings saying, "Yes, discriminate against gays!" If that happens, that (along with Citizens United and the crippling of the Voting Rights Act) will be his historical legacy; he'll be discussed in law schools right along with Roger Taney, as examples of now NOT to be a Chief Justice.
The arc of history still tends toward justice; it's just a bit longer than we had hoped. We will overcome this, too (barring nuclear war, of course).
They know Trump isn't one of them but they've embraced him with a religious ferver.
You mention Catholics. As a Catholic myself, the Roman Catholic church in the USA simply disgusts me. I think it is in the process of going rogue on Pope Francis, knowing full well that with his gentle and merciful nature, Francis will never bring the hammer down on the Archbishop Chaputs of the church. Yes, he's appointed some more open minded bishops, archbishops and cardinals while notably snubbing some like Chaput. But, those more open minded bishops, archbishops and cardinals will find contempt and resistance from the clerics and many churchgoers they lead. Chaput himself recently said the church would be better off if it was smaller and if non-traditional Catholics went away, pretty much a direct repudiation of Francis's mindset.
Truth is, throughout its nearly 2000 year history, the Catholic Church has always adapted to places it has gone to. For example, Christmas is a repackaging of the Celtic festival of lights. Down in Mexico, so many hallowed Catholic religious festivals such as the Day of the Dead are reinventions of millenia-old Native American festivals. Here in the USA, the Catholic Church long sought to emulate mainline Protestant Churches and encouraged Catholics to go out of their way to assimilate into public life and, like mainline Protestant churchgoers, piously observe the separation of church and state. However, with the steep decline of the mainline Protestant denominations and the rise of evangelicalism, the Catholic Church has become more evangelical and fundamentalist, demanding that its flock stand out from the dominant culture, fight the culture war and, if one of them runs for office, wear his/her faith on his/her sleeve. While the American Catholic Church doesn't subscribe to Biblical fundamentalism or teach that the Bible is the literal word of God or that the human race started with Adam and Eve or that some of the Old Testament patriarchs lived hundreds of years, etc., it subscribes to old time Catholic teaching which holds that sexual sin is the greatest evil out there and backburners if not minimizes social sin such as poverty, war and greed. Roman Catholicism in the US is quickly becoming a hostile isolated island in the global church. I would not be surprised if you start hearing dominant conservative Catholics actively and openly say "Francis is not my Pope."