This is false. Republicans don't always lie. If they did, literally nobody would believe them.
What they do is actually far worse. They tell just enough truth to make their lies seem believable. That's why they end up being dangerous: because they're believable.
I know you don't believe them or find them believable, but enough people do that they keep getting elected.
My solution to the pronoun game has always been to not worry about it.
At some point, the person having non-standard pronouns made a decision to have their pronouns not match their physical appearance, so it's up to them to communicate that difference in some other way. If they fail to do that adequately, there will be misunderstandings. Sometimes, that means they have to straight-up tell people when they meet them, other times it might mean a correction when a mistake is made. I've seen people wear buttons at social events, even, and I thought that was a cute solution.
If they want to be a dick about it, I now know that they're not someone I particularly want to be around anyway.