This really tells us a lot about how important email is to your business (ie not a lot).
I imagine your email server will work fine if you move all the email off it! However, that isn't really a solution to your problems - PST files are not very efficient, prone to corruption, insecure, and size-limited. I would imagine that running in this configuration will cause increasing levels of failure and support issues.
If an email is business critical, then it should be on the Exchange server, being regularly backed up, held efficiently and securely, protected against corruption by transaction logfiles, and being error-checked nightly as part of the online exchange-aware backup. If your server isn't big enough to handle all the business-critical email, then your business either needs to buy a bigger, more capable server, or it needs to move some of the roles away from it (as you have already done with the backup software - by the way, I hope the backup files weren't on the same disk volumes as the Exchange databases or transaction log files?). It's never a good idea to share an Exchange server with other infrastructure roles, it just compromises the performance and/or integrity of your email service.