On a Mac. operating system is Golden Gate 27.0. I ran an update to InDesign 21.6, and now when I open files, make changes, and try to save them, InDesign tells me that my files are damaged

“Dot-oh” versions are alway risky, whether it’s an OS or Adobe software. On a production machine, you should wait about 6-9 months before updating unless there is a feature you absolutely need. 

InDesign (as well as other Adobe products) is not supported on macOS 27 yet. In addition to this, I think I just saw reports that updating to InDesign 21.6 causes the problem you describe when saving files on a network.

So first, try to roll back to InDesign 21.5.x: Creative Cloud app > InDesign > 3-dot menu > Other Versions

If it doesn’t solve the issue, you’ll need to go back to a stable macOS version such as Tahoe. I don’t know if you updated your existing system to Golden Gate (in which case it was a premature step) or if this is a new machine with Golden Gate pre-installed. In any case, create a separate volume on your hard drive using Disk Utility, install Tahoe there, and work from there until macOS 27 is supported.