Most likely your Edge is busted or infected with virus/malware. First thing to do is to
MAKE SURE YOU BACKUP ALL YOUR PERSONAL DATA (LIBRARY FOLDERS) TO EXTERNAL MEDIA BEFORE ATTEMPTING TO FIX IT!
Many viruses attack both the built-in Browsers made by Microsoft. In W10, that's now Edge and IE11. Have you tested your ability to save favorites in IE11? If not, you probably have one of the new designer viruses that go after specifically MS browsers (Edge & IE11). To remove, use whatever your primary AV program installed is on your computer, scan/remove all viruses found. Next, go to
malwarebytes.org and download/install the free
MALWAREBYTES antispyware virus scanner. Scan/remove all found spyware viruses. Reboot your computer, and retest both your Edge and IE11, if IE11 now works but Edge doesn't, it's a more stubborn virus and you'll need more advanced tools to fix. If however, your Edge now saves favorites correctly, you're fixed and all set to go!!
If this doesn't fix your Edge, try using the built-in
SYSTEM RESTORE utility to rollback your window to a point in time when Edge was saving favorites correctly; a couple of weeks or even months might be required to do this. After restore reboots your computer; retest. If Edge now saves favorites correctly, your fixed and all set to go!
If none of these solutions worked, you may have a failed hardware component or simply need to reinstall your W10 using a W10 Clean Install. To perform your own hardware testing, try my Troubleshooting Guide I wrote here:
Windows 10 - Unclickable Task Bar
The Guide will produce a positive result if you follow it precisely and don't skip any steps about 90% of the time.
Post back if you're still stuck!
Happy Holidays!
<<<BIGBEARJEDI>>>