You don't say which version of OSX or OS you're using, but the problem appears to be at the operating system level. (I'm having no problems running these programs on my El Capitan, FWIW.)
Check out this possible solution.
Thanks for the reply. Mac OS 10.12.4 (I had put this in the tags). I have already used all the suggestions listed here and went to user chat for personal advice but it was basically the same thing - permissions, which I have covered. I was hoping to run into another Mac user who had encountered the same problem and might offer a solution. It is frustrating. I have been using Preimere Elements since version 8 and have never had this situation. I edit video for a community TV station and sort of depend on its stability. The frustration is never knowing when it will shut down with error 16. It always comes back for several weeks after a reinstall but will then suddenly quit. Yes, this has only happened with Sierra. Wish Adobe would be of more help.
Here's what worked for me with PS Elements 12 on MacOS 11.6 El Capitan. I keep getting the annoying Error 12 about once a year.
In Finder, navigate to /Library/Application Support/Adobe/
There should be a subfolder named "SLStore" that contains two files: "Adobe01.bmp" and one that is eight digits with file type ".lic"
Rename this folder and create a new empty ( ! ) folder named "SLStore". Set permissions for this folder as instructed.
Launch PSE. You should now be able to register with your Adobe ID and enter you license key again. If not, quit and launch again.