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Hi Jacsdad,
Thanks for clarifying your situation, so currently you have the following 2 issues.
1.For some reasons, the error indicator will suddenly show up on some of your cells which are formatted as Zip Code.
To proceed on this issue, we would need a copy of your sample workbook, in order to protect your personal information, we don’t recommend sharing your OneDrive link in the thread, please send this sample workbook to us via Private Message.
2.When you uncheck Turn on background error checking in Excel's preference setting, it would be automatically rechecked after restarting your Mac.
As Jim also mentioned in his reply, this issue could be caused by the corrupted user profile, please refer to Create a new user profile to create a new user on your Mac, check if the changes applied to Excel preference settings could stick under this new user profile. Let us know the result.
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Since this case has been last for a long time, please provide us with your current MacOS version and your Office version.
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Thanks for your cooperation.
Regards,
Alex Chen