I have also tried making a Page Rule to redirect www to bare, but still goes to WPEngine. I removed www records, page loading then times out (as expected). Added the records back, and STILL going to the old place destination. I have tried from multiple browsers, multiple networks, multiple machines.
May I ask have you asked WPEngine to check just in case if there is some CNAME record / hostname for your domain name at their side left?
It knows to be an issue when we use some hosting providers which enable SaaS for our domain name/website.
Kindly, ask them to re-check this and remove/delete if so.
I’d suggest you to follow the instructions from the article below as they are related to any SaaS integrator:
Therefore, just in case, kindly may I ask you to please re-check if the A www or CNAME www is existing under the DNS tab of Cloudflare dashboard for your domain name and if so, does it point to your new hosting / server IP address?