I don't think senators should be by state, I think senators should hold office for 5 years and every year the entire country should elect 20 senators.
Other things we should do:
Abolish political parties.
Uncap the house, algorithmically determine representative districts with something like the shortest split-line method, and assign between 3 and 5 representatives per district.
Break the powers of the president into multiple different offices.
Make the leaders of the house and senate elected offices.
I'm very curious to know how exactly you want to word this law to acheive the effect you're dreaming of without it being unenforceable, without it being weaponizeable as a mass voter suppression tool, and without creating a freedom of speech or freedom of assembly violation.
A fair voting system allows people to vote for whatever reason they want. Voters want to win. Banding together to focus and force multiply campaign resources increases chances to win. Political parties are an inevitability in a fair system.
I understand the vibe of your sentiment is to not allow political parties to grow to the overcentralizing control they have today. You're not particularly concerned about, say, a band of guys who meet up at the pub to figure out who they're gonna organize a collective vote for. At least I hope not, because the alternative sounds wildly dystopian. But like, what's the line in the sand between the two? How do you define the difference, legally?
The goal is to make it so self-declared private organizations can't be an official part of the election process. At the moment, the state holds primaries for political parties, and helps them keep track of who's in which one which helps maintain the duopoly.
I'm sympathetic to the concept, but I think that the advantages that organized parties have in terms of coordination (e.g. people with broadly similar values and policy goals choosing one candidate to represent those goals to avoid splitting the vote and seeing someone antithetical to those shared goals elected) are sufficiently strong that you would just see the current primaries replaced immediately by a primary process run completely independent of government oversight and resources. I can't imagine that being good from a perspective of electoral legitimacy or reducing the influence of money in politics.
Break the powers of the president into multiple different offices.
As long as we're talking esoteric political ideas, the big one here is to split head of state from head of government. It might not affect the function of government much, because the head of state is largely ceremonial in modern systems, but it's I think it's super-important psychologically.
A lot of (most?) people have trouble thinking about the office of the President as an abstract concept separately from the person of the President. Therefore, the President becomes an avatar of the United States, taken to be the living embodiment of our identity as a nation. That's why so many people freak out about "the destruction of America" when a member of the other party, with values they don't share, becomes the President, and it makes elections feel like a polarizing, existential referendum.
By contrast, King Charles is the head of state in the UK, while the head of government (the prime minister) comes and goes, and a stable avatar of the nation, largely above politics. They have their share of major problems over there, to be sure, but at least the nation has a shared identity to rally around when needed.
You still have plenty of people who are anti-monarchy in the UK. We also all know that the king is only a figurehead. It's not really a great solution to be honest.
Hey now, we don't want an actual democracy now do we? Think of the corporations. With all these broken up powers it's going to get really expensive to bribe them all to subvert the will of the people.