Election by lottery. Every eligible citizen is entered into the lottery to serve in the Senate. Each year on December 1st, 100 citizens are randomly selected to serve a 10-year term, to report to their duty by January 1st. The Senate will consist of 1000 of these people, with 10% of them replaced every year. (If a Senator dies, retires, or otherwise becomes unfit for office during their term, they are not replaced until their term ends, and the Senate will be slightly smaller than 1000.) The first year and the last year of your service in the Senate are special. During your fist year, you will be mentored by a more experienced Senator; you will have full voting rights, but you will be "Silenced" -- not able to propose new bills/votes or address the Senate -- until your second year. During your last year, you must choose at least one new Senator to mentor.

If selected, service in the Senate is considered mandatory, like being selected for a military draft. Individual Senators can be excused from service or allowed to retire before the end of their 10-year term, but only with approval from a 66% supermajority vote of the Senate. (This is important to help avoid selection bias in the Senate. Otherwise, certain demographics may be underrepresented if that demographic is more likely to refuse service or retire early. It's also important because the people who don't want to be in government are exactly who we need in government. If citizens are allowed to refuse service in the Senate, that would bias selection toward the type of people who want power, which may defeat the entire purpose.) Being a Senator should still be a prestigious, respected, and well-paid position, of course -- that will only further encourage people to accept their selection if chosen.

If, at any time or for any reason, the Senate has less than 500 Senators, a special selection will be performed immediately, and enough random citizens will be selected to bring the total number of Senators back up to 1000. (This includes the very first selection, since you'd be starting with 0 Senators, which is less than 500.) Since 100 new Senators must be placed every year, replacing the longest-serving ones, some of these Senators chosen in 'special selections' may end up serving terms of less than 10 years. (For example, in the very first year of the Senate, 10% of the Senators chosen will only serve for one year.)

The Senate can vote to "Silence" individual Senators -- to prevent them from proposing new bills/votes and prevent them from speaking to address the Senate, with a 66% supermajority vote. This can be a temporary punishment or a permanent injunction for the remainder of that Senator's term, at the Senate's discretion. Must have an individual vote for each individual Senator to be Silenced -- you can't Silence entire voting blocs with a single vote. Silenced senators, though, will still be in the Senate and will still vote just like any other Senator.

The Senate may (optionally) appoint a Chief Officer, who serves as the head of the Executive Branch, at the Senate's pleasure. The Chief Officer (and indeed any official in the government outside of the Senate and Judicial Branch) can be removed and replaced at the Senate's discretion at any time, with a simple majority vote. This Chief Officer can be a Senator, but doesn't have to be. Doesn't even have to be a citizen. It can be literally anyone the Senate agrees on (except for any previous Judge or Commander in Chief). Or, if the Senate chooses, they can have no Chief Officer at all, and instead have the heads of each individual department of the government report directly to the Senate. In the event of an exactly tied vote in the Senate (which will probably be very rare), the Chief Officer (if one exists) may cast the tie-breaking vote.

The Senate may (optionally) appoint a Commander in Chief of the Armed Forces, only if the Senate has declared war on another country. They may choose to appoint any General or Admiral who is currently serving or has previously served in the military to this role. The Senate will delegate control of the country's military to the Commander in Chief, who may make sweeping and rapid decisions about the military without consulting the Senate, on the presumption that their experience and ability to respond quickly will enable them to achieve better military results than the Senate or Chief Officer could. The Commander in Chief may only serve during times of declared war, and must step down from their position at the end of the war, ceding control of the military back to the Senate (optionally through the Chief Officer). Like the Chief Officer, the Commander in Chief can be removed and/or replaced at any time with a simple majority vote from the Senate, even if the war is still ongoing. No single person can serve as both Chief Officer and Commander in Chief -- not concurrently nor successively.

The Senate will also vote to appoint judges to the higher courts, including the Supreme Court, with a 57% vote required for successful appointment. They may only nominate well-qualified and experienced judges who are well versed in the law and the (new) constitution. These judges will serve for 20-year terms, after which they must retire and may never again serve in government in any other capacity. They can only be removed and replaced with a 75% supermajority vote from the Senate. (To prevent a constitutionally misguided Senate from quickly and easily replacing any judge that rules their new law to be unconstitutional.) And their most important job (as is supposedly the case already) is to rule on whether or not laws passed by the Senate and actions taken by the Executive Branch are allowable and enforceable according to the (new) constitution.

This ensures:

• The Senate always consists (mostly) of regular people with no political background, and people who are not particularly associated with or beholden to the ultra-wealthy.
• Some institutional knowledge and experience is built up and retained -- at any point, ~10% of the Senate will have 9+ years of experience in government.
• There will be reduced motivation for 'lobbying' and bribery. Since nobody is running for reelection and election campaigns as a whole are a thing of the past, Senators have no need to seek out donations and campaign contributions. They will not be concerned about their voting record having impacts on future campaign contributions. (And if any of them are greedy, well, the Senate is in full control of Senator's salaries. If they want more money, they can just vote to give themselves more money. No need to seek out bribes.) There's also no need for the society as a whole to go through the effort and expense of election campaigns anymore -- now they can put this effort and money into more useful causes.
• Nobody can have a lifetime career as a Senator. Every Senator will come from a non-political background and will probably be expecting to return to a non-political life after their 10-year term. They will (generally) understand the struggles and perspectives of common people, and (unless they're very old when selected) they will expect to have to live as a common person under the laws they passed once their term ends -- if they vote to pass a terrible law, they'll have to experience the consequences of that terrible law later.
• As the saying goes, "Those who want power the most are the ones you least want to have power." In this system, the Senate will be full of people who never wanted or asked for power, and the people who want and seek power won't be able to get it by running for the Senate, at least, because there's no way to try to be a Senator -- it's all just random luck. (Though they could still seek out positions in the Judicial Branch, the Executive Branch, or private industry. In the first two cases, it would be up to the Senate's judgement to prevent them from gaining that power.)
• The Senate will always be (approximately, statistically) representative of the people. Generally, all demographics and ideologies of the citizenry will be proportionally represented. (If the country is 50% female, you can expect ~500 Senators to be female. If the country is 7.2% gay, you can expect ~72 Senators to be gay. If 12% of the country is black, you can expect ~120 Senators to be black. If 20% of the population is atheist, you can expect ~200 Senators to be atheists. Etc.)
• No powerful "President" or "Prime Minister" or anything will exist. The Chief Officer (if present at all) is only there to be a manager on behalf of the Senate, to handle the day-to-day mundanities of running government services and/or respond to urgent situations that are developing too quickly to organize a vote in the Senate. The Chief Officer's power is inherently fragile and limited, because they can be replaced quickly and easily at any time if they displease a majority of the Senate.