Immigration generally requires that you be off-Island while a Full Work Permit (FWP) or Temporary Work Permit application is in process. However, this does not apply if you are already here on the island as a dependant on your spouse's work permit, or if you have an existing work permit in place with the same employer.
Do not book your flight until your permit has been approved or you will risk being asked to leave again and put on the next flight back out of Cayman. Further, although the vast majority of applications are approved without issue, there can be no guarantee that any application will be ultimately successful.
However, if you are already on-Island and are switching from one employer to another you may not need to leave the Island while your new work permit is processing. Having said this, you should make sure that your release letter from your current employer is dated to take effect on the exact date on which you hope to be authorised to commence gainful occupation in the service of your new employer. If this is not possible then you must get a Visitor's stamp in your passport on the day that your current work permit is cancelled, to avoid any prospect of you being labelled as an overstayer.