Shared primary ballots is a good start, but it seems like the most meaningful change to our political system would be implementing a system of ranked voting (https://en.wikipedia.org/wiki/Ranked_voting). Among other things, that would help break the two-party nightmare cased by first-past-the-post voting.
http://bcn.boulder.co.us/~neal/elections/approval_voting_lwv...
St. Louis in November passed approval voting. ( https://electionscience.org/press-releases/diverse-coalition... )
I think the lies and conspiracies are just a manifestation or side effect of the social and political divide. Its like imagining someone you already hate. "They're plotting against me", "they're going to get me", "they probably spilled that milk on purpose to piss me off", etc. And I think there is some substance to the hate, not to the conspiracy, but I see that our society has gotten to this point for real reasons and disagreements that can't be compromised on
edit: typos
A major issue seems to be literal lies and disinformation spreading over social media which makes that divide appear to be much greater than it actually is. If these blatant lies can be prevented then the left and right will have a much easier time coming to political compromises when running the country. Having shared primary ballots would also help a lot to prevent the most radical wings of each party from dominating, because then each party’s candidate would have to appeal to all voters rather than their own.