best UX testing method for a nonprofit site?

I work for a nonprofit and have been learning about UX and its methodologies but am no means an expert. We have a website that probably needs revamping and we want to get a sense right now of what a user takes away from the site when they visit - what do they think our mission is? where do we work? what types of people do we offer help to? etc...so far the resources I've found about UX testing seem to focus on products/apps/commercial websites. Are there good resources out there for UX specific to nonprofit sites? Is that an unnecessary separation, and the advice I see for commercial sites is also relevant to me?

I had been considering a moderated in-person test where participants think aloud as they navigate the site trying to answer a set of questions (like the ones I posed above about what the mission is, etc) while I take notes silently, but I would love feedback from experts on what might be the best method of eliciting accurate responses.

Thank you in advance!!