I didn't say that would be the only problem, but it is the only one I mentioned. Zimbra has a "Secure interprocess communication" option which is on by default. You can turn that off and then the certificates will work for internal communication regardless. Obviously that's not the best choice for being security conscious.
But yes, you can wait until close to June 8 to redo the certificates. As long as you get it done before that expiration date actually hits you'll be fine.
When you're using a self-signed certificate like that, your users will get a warning about the certificate no longer matching the first time they visit after you change it, and will have to store a new exception for the new certificate. The only way to avoid that is to get a commercial cert (which usually costs money).