I'd like to partner with organizations and companies that could use it (could be a great part of a larger teaching tool, or an ad-hoc backend for something like codepen) but have no idea how to go about finding them, other than cold outreach.
If you have a free product, how do you give it away? I know this sounds a little like "What marketing?" but are there any channels or strategies that work particularly well when price isn't an issue?
I've had success with cold outreach to users on Reddit (random DMs to them) and Discord communities. Now neither of these are going to help you (unless you've also got a video game product!) but I think it demonstrates that you need to go where the users are and not be afraid to tell people about it. If the product is genuinely useful to the target audience then you're helping them out by letting them know about it, the is especially true if it's free.
Finally, I try to think of marketing like I do with programming. You're solving problems and the solutions to those problems don't always come to you straight away. Sometimes you think of the solution when you're in the shower or on a walk. Sometimes there's more than one solution, or one that works much better than another but you only discover it later. The important part is to keep thinking about the problem and to keep turning it over in your head.
That means that even though it's free, you have to sell it - maybe that's just through marketing if it's simple enough, or maybe you're going to have to get on the phone with decision makers if it's more complex.
If it's the latter, the fact that it's free honestly may not even be a good thing. Companies have budgets and willingness to pay for things, so the person evaluating it may prefer something that's paid because there's money available anyway and a paid tool is much less likely to disappear tomorrow.
I really recommend thinking about pricing and trying to sell it to a few customers - if that's your ultimate goal anyway, it means you'll be getting feedback from the right people (those who are willing to evaluate it as a paid tool) vs. folks who may only be willing to use it because it's free.
I'd actually argue you shouldn't offer anything for free, but instead charge whatever makes sense factoring in you're going to have to support customers even if it was a free product. For example if your offering website uptime monitoring, but it's not accurate that can cause a real cost to me