I'm not a cPanel or Plesk expert, but I've dabbled in both -- along with Webmin. If Plesk should happen to include a migration tool for your situation, that's what you will want, and even then you'll need to proceed with caution. (I have no idea whether one exists, BTW.)
The thing to keep in mind is these web-based front ends (cPanel, Plesk, et al.) mangle config files on your server into their own specific, pseudo-proprietary* format, and if you or another process comes along and mucks with those, things start to break.
The "quick", "safe" way is to grow your available capacity without migrating. If your hosting provider doesn't offer that capability, the next best thing would be to order a capacity upgrade that they support. If you insist on changing hosting providers and front end management utilities, then be prepared to take backups and start from scratch.
* It's still text-based, and it is still more or less readable, but their parser expects configs to be in a specific place and in a specific format.
Last edited by anomie; 04-09-2011 at 07:30 PM.