I demonstrated Mailpoet to you guys and it has a free version, but it does not allow you to segment user lists. So if you wanted to separate expired members from active, or patron from adult, you’d not be able on the free plan. From there it goes up to Monthly 10 dollars.
On further research I found this one which also integrates – Mailster – for 1 time fee of 89 dollars. Mailster appears to do more than Mailpoet, but doesn’t have a free level if less is fine.
Of course you always have your Mailchimp, but that won’t integrate into the site.