What Options Exist For U.S. Citizens Looking To Move To Canada?

Relatively few, at the moment. If you're a skilled worker in demand - which could be anything from a welder to a doctor to (questionably, but factually) an executive - you can try to immigrate economically.
If you figure you can find work under the table, just crossing the border and not crossing back would be doable. The refugee path is currently closed by treaty, unfortunately. Hopefully that will change.
- My background is Chemical Engineering, at one of the largest tire manufacturers. Hopefully with several recent regional and global awards under my belt, that might help.
  My wife is a dietitian and I swear she would be in more demand than I am, but she doesn’t want to do the applying. Rather she wants to have me find something and her figure it out upon moving, if that were to happen.
  
  It used to be that there was a website where you could basically just check your eligibility, but the system got somehow even more complicated so checking "express" eligibility is the best I can do: https://www.canada.ca/en/immigration-refugees-citizenship/services/immigrate-canada/express-entry/check-score.html
  Generally speaking, the highest-scoring applicants get in first. If that fails, maybe there's some niche provincial program or something, but you'll need to hire an immigration lawyer to have a chance of figuring it out.
  If you do get let in, I recommend driving up in an RV. The housing market in Canada is still really fucked, and that's a decent fallback option. Winter-safe ones exist, but I also see people building little insulated enclosures, and in any case cold season is nearly over right now.

I can't help with immigration but your replies moved me. I know how it looks in America right now. I understand you feel your life is in danger and I don't even disagree.
I just wanted to let you know there are more people on your side than you might realize.
I'm a redneck cowboy. I grew up on a farm in Texas we raised horses. I live in Florida now working at a western wear store and a.dollar branded store.
I wear cowboy boots, I carry a 1911 kimber .45 all day every day hooked into the leather belt I made myself with a big brass belt buckle that says "Marlboro". A promotional item I got back in the day by trading in my points.
If you saw me walk past you at a grocery store you'd think "there's another trumpet, there's another one who wants people like me dead"
My point is that isn't true. Id gladly give my life to defend your rights and I'd be proud to do it.
My family has been here since the late 1600s. We fought the British, we fought slavers, we fought the Nazis, my grandad even helped eliminate the rabies problem that was going around.
This is my country and I'll be damned if let anyone tell you it ain't your country just as much as it is mine. Ill fight to the death to make sure it always is your country too. I can't be everywhere, but we are everywhere. You might not see people like me when we walk by you. I promise you are there seeing what some folk are trying to do to you and we are very pissed off as well.
My grandad killed the Nazis once. I'll kill them again if I have to. We are there and we are also worried about your safety. I'm pissed to. If pushing leads to shoveing, well, my friend I am a damn good shot with my gun.
I'm sorry my country, you're country, has made you feel unwelcomed. I'm ashamed of that.
That's all I wanted to say.

Family sponsorship
Work sponsorship (requires education, skills, and/or experience)
A disgusting amount of money
- Guess it’s the ol’ #2.

Unfortunately it’s easier said than done. They clamped down on immigration a few years back (because they were letting in a bunch of people while also not doing anything about the housing market and got understandable blowback).
Also I’ll point out that Canada is only one (goose) step behind the USA, and may not stay safe. There are a lot of fascists and bigots here too.
- I understand, and Canada hasn’t been the only country in perusing.
  I can hope that the emboldened nature of US fascists helps wake up other countries that may not have fallen too far already.
  
  I hope so too, though the leadup to ww2 doesn’t give me a lot of hope, since domestic fascism remained popular in allied countries up until fighting the axis

lookup CUSMA visa.. Equivalent of TN for Canadians. If your profession falls into covered category all you need a job offer at the border. You could come over on tourist visa, scout for work, get offer, go back to the border, stamp visa in and enjoy Canada

https://www.canada.ca/en/immigration-refugees-citizenship/services/immigrate-canada/express-entry.html

Renounce your US citizenship at the border while holding a Beaver and Canadian Goose, screaming "Oh Canada" at the top of your lungs until someone lets you in.
- I actually do know the majority of “Oh Canada”.
  Goose, easy.
  Beaver? I’m fucked.
  Will maple syrup make do?
  
  You'll have to be covered in maple syrup, we took that as a given
  
  Goose is easy? Geese are assholes good luck at coming close enough to carry a living goose.

Just to head the idea off at the pass: there is no material version of "90 day fiance" permanent residency for Canada.
- My wife would have issues with that regardless. 🥴

Also, do you have family in Canada? If so, family reunification might be an option.
Also look into CUSMA work permit or a US intracorporate transfer if you are working in a company with a canadian branch
Otherwise, look into PNP programs
- No family in Canada. Most are among MAGA in the states.
  The company I work at isn’t hiring at its Canadian location.
  I’ve been applying to jobs that are applicable to my background and experience, but I know that requiring sponsorship is going to hinder that progress.
  While I’m fluent in English, my French is subpar to say the least. Being mid-30’s isn’t helping my score regarding express entry either.
  I’m not expecting anything to happen overnight.
  
  You don't "need" to speak French unless you come to Quebec and even then, if you do you will have access to some classes. Some companies offer them to theit expat on site and there's some from the government.
  But you don't have to worry about that if you move to litterally any of the other provinces.

Why don't you stay there and fix your country?
- Personally? My country wants me dead. "Oh but claim refugee status at the border" yeah, that don't work if you're coming from the US. The US is considered a safe harbor by Canada for LGBT people.
  Let that sink in a moment. Canada still considers the US to be a safe harbor for people who are LGBT, and so explicitly (unless something changed literally in the past couple months) blocks refugees from the US.
  
  it's going to be increasingly hard for CA to justify US as a safe harbor for LGBT people when the US has literally erased the T, and is starting even to block trans people from obtaining US visas. not that I'm terribly optimistic that CA will even bother justifying it.
  
  I'm doing what i can to advocate to my politicians to get that changed
- After years of putting oneself in danger and doing rather than saying, you start to look at how can you improve the life of your family. The battle of trying to make this country better, is it what is best for my family?