Fix email deliverability issues on WHM / make it work with my SendGrid account

I need to fix my WHM/cPanel DNS Propagation and Email Deliverability Issues on Domains hosted on my server / make it work with SendGrid. I used to have SendGrid working just fine but started running into deliverability issues which are related to the sending IP of my server, and was unable to make the server send through a different IP (which the server has several IP's allocated to it), using the SendGrid/Exim configuration.