I'm not an American and unable to directly participate, so my own thoughts are perhaps no more helpful than yours. Nonetheless, I think there are a couple of things to bear in mind.
- It is worth remembering that the "simmering hatred" being perceived is in large part (though not wholly) a product of online interactions and discussions. There are communities and situations where that hostility is real, direct and personal against other people personally known by their real names and faces, but they are less numerous and widespread than one might expect going by the 'Net. Therefore, one action that might help minimize this effect would be to try to keep one's political interactions, where possible, on the local and personal level. Get involved in local school boards and town halls. If nothing else, this makes it clear exactly how opinions are really distributed.
- Try to understand what an antagonistic individual or group really wants by whatever they're declaring their "victory conditions": if what they ultimately want is protections for a particular group, can those protections be achieved by a solution you can both live with, if it's not their preferred one? It may be possible to come to productive compromises that way. (This is not always practical because people are not always honest about this, either to opponents, allies, or even themselves, but if compromise is possible this is usually a critical step.)
- Be willing to accept that compromise is the art of managing disappointment. This is one point where I see, perhaps, the Left being a little too stubborn about rejecting this and the Right being a little too quick to accept it, which is one reason the conflict has continued without finding productive resolution either way.