All organizations will need email marketing to send broadcast newsletters. You will also need email marketing automation, at least basic automation like autoresponders (drip campaigns). Some email marketing providers also provide more advanced automation like tagging, segmenting, conditional blocks, etc.
MailGet Bolt has really good pricing. You can start for free to send 9,000 emails to 300 subscribers. For $29/month you can send unlimited emails to 5,000 subscribers. MailGet Bolt already includes built-in SMTP server but you can add your own SMTP servers like Mailgun, SendGrid, or AWS SES.
Moosend has advanced automation features. Using the free plan, you can send emails up to 1,000 subscribers.
EmailOctopus gives you 2,500 subscribers for free and you can send up to 10,000 emails per month. Make sure to apply for the 20% nonprofit discount.
Mailjet free plan provides you with 6,000 emails per month, sent at 200 a rate of emails per day.