The fundamental cause of America’s problem is the two-party system. If you want to get rid of that you have to switch to a proportional representation system. I would suggest working at the local or state level. I do not know of any organization working on this issue. You would likely have to start one yourself or hire someone else to do so.
If you’re genuinely going to do it, any suggestion I make here about specifics would be pointless, as you should do significant research before deciding on what flavor of proportional representation to push and where. But, the key is to adopt a system known for accurate and small party representation. If a party gets enough votes to win a single seat, they should be awarded a single seat. If they get a third of the votes, they should get a third of the seats.