You probably should be asking yourself how important is this to you/your company and how is it going to be resourced, because the operational overhead of running this is non-trivial, and certainly not set-and-forget. You technically can do it yourself, but there's a reason ESPs[1] exist.
[1] Email Service Provider, before someone complains.
Disclaimer: I used to work for one of the big ESPs, but not for any of the ones I'll mention. My old company is more on the enterprise side these days. Not an endorsement of anyone and all that...
Free? You get what you pay for. Like I said, consider how much is this worth to your business. Email is hard work.
I know a lot of people who use Mailchimp for this and seem to be generally happy with them. Also Constant Contact and HubSpot (the latter I know has a free tier of some sort). There are a lot of players these days, many new since I was in the business, and a search for "Email Service Provider Newsletter" will turn up a ton.
[1] Email Service Provider, before someone complains.