Boot up and hold down Command-R.http://www.idownloadblog.com/2016/02/24/how-to-start-mac-recovery-mode/
Run Disk Utility first from there to check for drive errors. Might as well Repair Permissions too. After that quit Disk Utility and it will take you back to the 4 choices window.
One of the choices will be to Reinstall OS X, you can reinstall without losing any data, it will just reinstall the OS. Since you did an Upgrade I don't know if it will reinstall Sierra, or what you had on it before. BTW it will take a while as it will need to re-download the installer.
If you have a Time Machine backup you can choose to Restore from it to get you get you back to where you were before the Upgrade. It will take a while also as it has to rebuild the drive.
Can't afford to lose my backups
That part I don't understand though. If you're talking about what's on the HD now that isn't a backup, that's the original source data. If you're talking a Time Machine Backup you won't lose that, that's a separate drive.
Good luck, I'll keep an eye out here.