For starters, it sounds like you need to spend more time really nailing down what kind of work you want to do. Do you want to continue doing research, or are you interested in more of a program- or policy-related role? Start by identifying some of the kinds of work you want to do, then set up informational interviews with people who are in those roles. That should give you some ideas and leads; it might even generate some part-time work that could help you get your foot in the door. As with anything, it's about the hustle and the network. You're not likely to get a way in to the kind of position you want just by trawling ads on Indeed.
FWIW, I'm a PhD working for a nonprofit, married to a PhD who is trying to transition from postdocs into a nonprofit. I think we exemplify 2 different ways in:
--I took a job that required just a BA. I already had connections with the org. from previous part-time/temp work. For me, it has been a great decision, and I'm working my way into more advanced roles.
--My spouse is working as a consultant while trying to get hired by another NGO. Openings are few and far between. He's applying for everything, doing contract work on projects related to his field, and patiently building connections.
I'll second Hawkeye--you really have to show an organization that you understand their mission and can contribute to the work they need done.
One of the biggest hurdles to leaving academia for the NGO world is being able to show transferable skills. Very few NGOs are interested in the kind of "advance the state of the field" science that you have likely been pursuing over the course of your career so far. For NGOs, it's almost all about applied science and management. PhDs sometimes assume that they should be able to jump right in to high-level positions, and they are often unaware of their experience gaps. You might have to revise your expectations about where you can start.
And a word on mindset. For better or for worse, academia trains us to poke holes in new ideas and approaches when others present them to us (hi reviewer 2, thanks for eviscerating the manuscript). Arguably, this is good for developing sound science, but it can be a barrier on the outside. It can come across as condescending and counterproductive. You seem to be denigrating management, but that stuff matters. When it comes to advancing policy or projects on the ground, fundraising, communications, and management are important. PhDs who have spend their careers in the academy sometimes don't get that. For example, an acquaintance leaving academia recently started an initiative to fund something related to his research. When a big interest group didn't jump at the chance immediately, he berated them on social media. That's, like, fundraising Don't #1. So try to keep an open mind. Some of those folks who seem to have "scant knowledge of science" might have other skills that make them really valuable. In my experience in the NGO world, scientific knowledge doesn't matter much if you can't use it to get things done.
As far as think tanks, I've got no clue. I can only assume that getting hired by one requires a certain level of family connections or a MacArthur fellowship.