Email Issues After Moving Domain to New Linode

I've had multiple domains hosted on an old existing Linode using Ubuntu 12.04. They're each running Wordpress instances. I'm using Godaddy as a third party email provider and I'm having no issues with emails from my websites' contact forms.

I've just built a new Linode running Ubuntu 14.04 using the Getting Started guide (https://www.linode.com/docs/getting-started) so that I can transfer my sites to this new server. I've transferred DNS records for only one of the sites so far, but now emails from my contact forms are no longer working.

What tests should I run or files should I examine to investigate the issue. My PHP error logs aren't showing anything.

Some other details that might be pertinent:

1. Neither Linode has the SendMail package installed

2. I've manually changed the hostname for my new server whereas the old server has only its default name assigned by Linode

3. Instead of using the DNS Manager to delete the old domain zone and subsequently rebuild a new domain zone with the new IP address, I only edited the old domain zone changing the IP address's for the A records.