Working together takes effort. That has never been more true than today, when you can flip between two news channels covering the same events and come away with two completely different, and highly editorialized, perspectives on those events. It is more important than ever that each of us resists the temptation to withdraw into the news bubble with which we feel most comfortable. Rather, we should force ourselves to listen to the coverage that makes us uncomfortable. We should strive to set aside our emotion and understand the merits of the case being presented by this coverage. And there is almost always some merit to the other side's position.
I am not saying that we should settle when we believe important principles are at stake. I am saying we should realize that the people with whom we disagree also are fighting for principles important to them. The idea is not to avoid conflict at all cost--that is moral cowardice. Rather, we should seek a deeper understanding of issues to use as leverage to get past conflict to solutions that work for all of us.
None of us has a monopoly on being right. In fact, many of us are probably operating on completely different notions of what "being right" means. I can think of three different paradigms for "right" without too much effort: (1) "right" as in compliance with some deeply held, personal moral / religious belief; (2) "right" in the sense of compliance with the system of procedural justice defined by our Constitution; and (3) "right" in the pragmatic sense of being most likely to achieve some desired end. These three paradigms potentially conflict with each other, of course. Moreover, even people who agree on a paradigm for "right" can disagree over how to apply that paradigm in a particular case.
The authors of our Constitution clearly understood the different paradigms for "being right" and the potential for disagreement within each paradigm. They enshrined the freedom of conscience of each individual within the First Amendment to the Constitution. They created the judicial branch of government as a way to adjudicate differences in interpretation of the Constitution itself. And they were so completely unable to resolve the most pressing social issue of their day--slavery--that the best "pragmatic" solution they could come up with was to basically table the issue for 20 years. The fact that we disagree is not a sign that our republic is broken. It is a sign that our republic works as well as it has ever worked.
Our republic is in danger only when we stop talking to each other, when we game the system for partisan ends, and stop using the procedural mechanisms designed to help us resolve our differences.