CloudMagic is about to stop working on September 30th. Can I prolong its life?
There are two ways Cloudmagic could make their app stop working on a certain date:


The destruction date could be built in to late versions of the Mac client itself, checking on startup if it's September 30th yet or not.
The Cloudmagic server (yes, your email is on their servers, that's why they starting needing money to support it) could start checking to make sure every client it's giving data to is Newton and not Cloudmagic (on Sep 30): most likely Newton has a special verification key that Cloudmagic doesn't, which was added in the name-changing update.


Regardless of which of these routes the developers took, there's nothing you can do to prevent the built-in destruction date. 

Edit: You should note that updating to Newton will get you a 12-month free trial with no strings attached; I'm taking advantage of that right now.