I’d use Disk Utility or the Carbon Copy Cloner app and clone the whole device out to a scratch storage device.
Copying your home directory will get most of your stuff. It’ll be harder (and manual) to restore, though.
Time Machine would be typical, but that’d more I/O and take longer than might be appropriate here.
Particularly if the storage hardware is failing.
The hard drive hardware tests are Diagnostics and Hardware Test.
Those testing tools don’t detect all errors, but they do detect common failures.
Failing hard drives get slow and flaky.
File system verifications detect corrupt file systems, but not necessarily corrupt files.
And I’d not try any tests, nor first aid, nor any repairs, until after I had a couple of backup copies of my data.