All the software has the same problem - both Mac native apps (be sure to try TextEdit), and any third-party apps?
Or, is there some particular app that is your primary use?
Can you try to give more details about which apps those are?
You mentioned an exported file will show that 05:34. So, does that mean that a simple saved file will ALSO show that same time?
This would be an ideal time to create a new user, log in to that new account, then try your primary app to see if that time problem still exists, even in a different user account! If Time is still stuck, even in a different account, that would indicate a system-wide issue, and re-installing will be an even better choice at that point.
(I am not suggesting a wipe and install, just a reinstall of the system. The reinstall simply reinstalls (and verifies that system files are complete, not corrupted, and in the correct locations) Might only take about 30-45 minutes. This is the time to use a bootable USB installer for your system, which would be significantly faster than downloading through your recovery partition! )