I made a career pivot in 2019. I did the following in my cover letter:
1. Took a few of the bigger responsibilities and experience requirements, and called them out in the cover letter with an explanation of how my experience fit in. In my case, I was leaving the nonprofit world and going into a science/medical related field. So I talked about how my work at the nonprofit related to healthcare and gave me the ability to translate complex concepts into accessible terms. I talked about how I planned publications (even though the publications were like, fundraising brochures and annual reports, not medical communications) and managed planning and deadlines. Etc.
2. Remember that the letter should explain how you can help them-- they are really going to care much more about what you can do for them, than why you are changing fields. Are you looking to do communications type work for them, or are you trying to move into program management? Either way, if you were planning and managing marketing campaigns, there are a lot of project management skills that will translate directly that you should emphasize.
3. I would not reach out to anyone that you submitted a resume unless you already know or have a connection to that person.
Askamanager.com has really great advice and good examples of cover letters for a lot of different situations.
I have helped hire a good number of people for nonprofits, and even if the ad says "nonprofit experience preferred", we generally welcomed all similar experience.
I will use my 20 years of experience to give you one piece of advice when job hunting in the nonprofit world: I would strongly recommend avoiding, or at least treading very carefully before accepting, a job at an organization that is still run by the founder.
ETA: It also goes over well if you can add a line or two that talks about why their mission is meaningful to you.
I would like to wish you a good luck! Hope you will find your dream job soon. As for me, i never really had a job that i hate, not for a long at least. I was able to tell myself the truth and look for a better options. These days i am doing some freelance jobs and communicate with the clients with the microsoft teams phone number only. No personal meetings or calls, which makes my job really nice.